HOME

TheInfoList



Click Here for Items Related To - General topology
OR:
General topology on:   [Wikipedia]   [Google]   [Amazon]

In mathematics, general topology is the branch of
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
that deals with the basic
set-theoretic Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concer ...
definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology,
geometric topology In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another. History Geometric topology as an area distinct from algebraic topology may be said to have originated ...
, and
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify u ...
. Another name for general topology is point-set topology. The fundamental concepts in point-set topology are ''continuity'', ''compactness'', and ''connectedness'': *
Continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...
s, intuitively, take nearby points to nearby points. *
Compact set In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", ...
s are those that can be covered by finitely many sets of arbitrarily small size. *
Connected set In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties th ...
s are sets that cannot be divided into two pieces that are far apart. The terms 'nearby', 'arbitrarily small', and 'far apart' can all be made precise by using the concept of
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are ...
s. If we change the definition of 'open set', we change what continuous functions, compact sets, and connected sets are. Each choice of definition for 'open set' is called a ''topology''. A set with a topology is called a ''
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
''. ''
Metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
s'' are an important class of topological spaces where a real, non-negative distance, also called a ''
metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathema ...
'', can be defined on pairs of points in the set. Having a metric simplifies many proofs, and many of the most common topological spaces are metric spaces.


History

General topology grew out of a number of areas, most importantly the following: *the detailed study of subsets of the
real line In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a poin ...
(once known as the ''topology of point sets''; this usage is now obsolete) *the introduction of the
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...
concept *the study of
metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
s, especially
normed linear space In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" ...
s, in the early days of
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined ...
. General topology assumed its present form around 1940. It captures, one might say, almost everything in the intuition of continuity, in a technically adequate form that can be applied in any area of mathematics.


A topology on a set

Let ''X'' be a set and let ''τ'' be a
family Family (from la, familia) is a group of people related either by consanguinity (by recognized birth) or affinity (by marriage or other relationship). The purpose of the family is to maintain the well-being of its members and of society. Ideal ...
of
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
s of ''X''. Then ''τ'' is called a ''topology on X'' if: # Both the
empty set In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
and ''X'' are elements of ''τ'' # Any union of elements of ''τ'' is an element of ''τ'' # Any
intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their ...
of finitely many elements of ''τ'' is an element of ''τ'' If ''τ'' is a topology on ''X'', then the pair (''X'', ''τ'') is called a ''topological space''. The notation ''Xτ'' may be used to denote a set ''X'' endowed with the particular topology ''τ''. The members of ''τ'' are called ''
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are ...
s'' in ''X''. A subset of ''X'' is said to be
closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
if its complement is in ''τ'' (i.e., its complement is open). A subset of ''X'' may be open, closed, both (
clopen set In topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open and closed. That this is possible may seem counter-intuitive, as the common meanings of and are antonyms, but their mathematical d ...
), or neither. The empty set and ''X'' itself are always both closed and open.


Basis for a topology

A base (or basis) ''B'' for a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
''X'' with
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
''T'' is a collection of
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are ...
s in ''T'' such that every open set in ''T'' can be written as a union of elements of ''B''. We say that the base ''generates'' the topology ''T''. Bases are useful because many properties of topologies can be reduced to statements about a base that generates that topology—and because many topologies are most easily defined in terms of a base that generates them.


Subspace and quotient

Every subset of a topological space can be given the
subspace topology In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced t ...
in which the open sets are the intersections of the open sets of the larger space with the subset. For any
indexed family In mathematics, a family, or indexed family, is informally a collection of objects, each associated with an index from some index set. For example, a ''family of real numbers, indexed by the set of integers'' is a collection of real numbers, wh ...
of topological spaces, the product can be given the
product topology In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seemi ...
, which is generated by the inverse images of open sets of the factors under the projection mappings. For example, in finite products, a basis for the product topology consists of all products of open sets. For infinite products, there is the additional requirement that in a basic open set, all but finitely many of its projections are the entire space. A quotient space is defined as follows: if ''X'' is a topological space and ''Y'' is a set, and if ''f'' : ''X''→ ''Y'' is a
surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of ...
function, then the
quotient topology In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient t ...
on ''Y'' is the collection of subsets of ''Y'' that have open
inverse image In mathematics, the image of a function is the set of all output values it may produce. More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or through) ...
s under ''f''. In other words, the quotient topology is the finest topology on ''Y'' for which ''f'' is continuous. A common example of a quotient topology is when an
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relatio ...
is defined on the topological space ''X''. The map ''f'' is then the natural projection onto the set of
equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
es.


Examples of topological spaces

A given set may have many different topologies. If a set is given a different topology, it is viewed as a different topological space.


Discrete and trivial topologies

Any set can be given the discrete topology, in which every subset is open. The only convergent sequences or nets in this topology are those that are eventually constant. Also, any set can be given the
trivial topology In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the consequ ...
(also called the indiscrete topology), in which only the empty set and the whole space are open. Every sequence and net in this topology converges to every point of the space. This example shows that in general topological spaces, limits of sequences need not be unique. However, often topological spaces must be Hausdorff spaces where limit points are unique.


Cofinite and cocountable topologies

Any set can be given the cofinite topology in which the open sets are the empty set and the sets whose complement is finite. This is the smallest T1 topology on any infinite set. Any set can be given the cocountable topology, in which a set is defined as open if it is either empty or its complement is countable. When the set is uncountable, this topology serves as a counterexample in many situations.


Topologies on the real and complex numbers

There are many ways to define a topology on R, the set of
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Ever ...
s. The standard topology on R is generated by the open intervals. The set of all open intervals forms a base or basis for the topology, meaning that every open set is a union of some collection of sets from the base. In particular, this means that a set is open if there exists an open interval of non zero radius about every point in the set. More generally, the
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
s R''n'' can be given a topology. In the usual topology on R''n'' the basic open sets are the open balls. Similarly, C, the set of
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s, and C''n'' have a standard topology in which the basic open sets are open balls. The real line can also be given the
lower limit topology In mathematics, the lower limit topology or right half-open interval topology is a topology defined on the set \mathbb of real numbers; it is different from the standard topology on \mathbb (generated by the open intervals) and has a number of inte ...
. Here, the basic open sets are the half open intervals [''a'', ''b''). This topology on R is strictly finer than the Euclidean topology defined above; a sequence converges to a point in this topology if and only if it converges from above in the Euclidean topology. This example shows that a set may have many distinct topologies defined on it.


The metric topology

Every
metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
can be given a metric topology, in which the basic open sets are open balls defined by the metric. This is the standard topology on any normed vector space. On a finite-dimensional vector space this topology is the same for all norms.


Further examples

* There exist numerous topologies on any given finite set. Such spaces are called
finite topological space In mathematics, a finite topological space is a topological space for which the underlying point set is finite. That is, it is a topological space which has only finitely many elements. Finite topological spaces are often used to provide examples ...
s. Finite spaces are sometimes used to provide examples or counterexamples to conjectures about topological spaces in general. * Every
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...
has a
natural topology In any domain of mathematics, a space has a natural topology if there is a topology on the space which is "best adapted" to its study within the domain in question. In many cases this imprecise definition means little more than the assertion that ...
, since it is locally Euclidean. Similarly, every
simplex In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
and every
simplicial complex In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial s ...
inherits a natural topology from Rn. * The
Zariski topology In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is n ...
is defined algebraically on the
spectrum of a ring In commutative algebra, the prime spectrum (or simply the spectrum) of a ring ''R'' is the set of all prime ideals of ''R'', and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with th ...
or an
algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. M ...
. On R''n'' or C''n'', the closed sets of the Zariski topology are the solution sets of systems of
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An examp ...
equations. * A linear graph has a natural topology that generalises many of the geometric aspects of
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties * Graph (topology), a topological space resembling a graph in the sense of discr ...
s with vertices and edges. * Many sets of linear operators in
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined ...
are endowed with topologies that are defined by specifying when a particular sequence of functions converges to the zero function. * Any
local field In mathematics, a field ''K'' is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. Equivalently, a local field is a locally compact ...
has a topology native to it, and this can be extended to vector spaces over that field. * The
Sierpiński space In mathematics, the Sierpiński space (or the connected two-point set) is a finite topological space with two points, only one of which is closed. It is the smallest example of a topological space which is neither trivial nor discrete. It is named ...
is the simplest non-discrete topological space. It has important relations to the
theory of computation In theoretical computer science and mathematics, the theory of computation is the branch that deals with what problems can be solved on a model of computation, using an algorithm, how efficiently they can be solved or to what degree (e.g., ...
and semantics. * If Γ is an
ordinal number In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the least ...
, then the set Γ = , Γ)_may_be_endowed_with_the_order_topology_generated_by_the_intervals_(''a'', ''b''),_[0, ''b'')_and_(''a'', Γ)_where_''a''_and_''b''_are_elements_of_Γ.


_Continuous_functions

Continuity_is_expressed_in_terms_of_ , Γ)_may_be_endowed_with_the_order_topology_generated_by_the_intervals_(''a'', ''b''),_[0, ''b'')_and_(''a'', Γ)_where_''a''_and_''b''_are_elements_of_Γ.


_Continuous_functions

Continuity_is_expressed_in_terms_of_neighborhood_(topology)">neighborhood_ A_neighbourhood_(British_English,_Irish_English,_Australian_English_and_Canadian_English)_or_neighborhood_(American_English;__see_spelling_differences)_is_a_geographically_localised_community_within_a_larger_city,_town,_suburb_or_rural_area,__...
s:__is_continuous_at_some_point__if_and_only_if_for_any_neighborhood__of_,_there_is_a_neighborhood__of__such_that_._Intuitively,_continuity_means_no_matter_how_"small"__becomes,_there_is_always_a__containing__that_maps_inside__and_whose_image_under__contains_._This_is_equivalent_to_the_condition_that_the_Image_(mathematics)#Inverse_image.html" ;"title="neighborhood_(topology).html" "title="order_topology.html" ;"title=", Γ) may be endowed with the , Γ)_may_be_endowed_with_the_order_topology_generated_by_the_intervals_(''a'', ''b''),_[0, ''b'')_and_(''a'', Γ)_where_''a''_and_''b''_are_elements_of_Γ.


_Continuous_functions

Continuity_is_expressed_in_terms_of_neighborhood_(topology)">neighborhood_ A_neighbourhood_(British_English,_Irish_English,_Australian_English_and_Canadian_English)_or_neighborhood_(American_English;__see_spelling_differences)_is_a_geographically_localised_community_within_a_larger_city,_town,_suburb_or_rural_area,__...
s:__is_continuous_at_some_point__if_and_only_if_for_any_neighborhood__of_,_there_is_a_neighborhood__of__such_that_._Intuitively,_continuity_means_no_matter_how_"small"__becomes,_there_is_always_a__containing__that_maps_inside__and_whose_image_under__contains_._This_is_equivalent_to_the_condition_that_the_Image_(mathematics)#Inverse_image">preimages_of_the_open_(closed)_sets_in__are_open_(closed)_in_._In_metric_spaces,_this_definition_is_equivalent_to_the_epsilon-delta_definition.html" ;"title="order topology">, Γ) may be endowed with the order topology generated by the intervals (''a'', ''b''), [0, ''b'') and (''a'', Γ) where ''a'' and ''b'' are elements of Γ.


Continuous functions

Continuity is expressed in terms of neighborhood (topology)">neighborhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
s: is continuous at some point if and only if for any neighborhood of , there is a neighborhood of such that . Intuitively, continuity means no matter how "small" becomes, there is always a containing that maps inside and whose image under contains . This is equivalent to the condition that the Image (mathematics)#Inverse image">preimages of the open (closed) sets in are open (closed) in . In metric spaces, this definition is equivalent to the epsilon-delta definition">ε–δ-definition that is often used in analysis. An extreme example: if a set is given the discrete topology, all functions :f\colon X \rightarrow T to any topological space are continuous. On the other hand, if is equipped with the indiscrete topology and the space set is at least T0, then the only continuous functions are the constant functions. Conversely, any function whose range is indiscrete is continuous.


Alternative definitions

Several Characterizations of the category of topological spaces, equivalent definitions for a topological structure exist and thus there are several equivalent ways to define a continuous function.


Neighborhood definition

Definitions based on preimages are often difficult to use directly. The following criterion expresses continuity in terms of
neighborhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
s: ''f'' is continuous at some point ''x'' ∈ ''X'' if and only if for any neighborhood ''V'' of ''f''(''x''), there is a neighborhood ''U'' of ''x'' such that ''f''(''U'') ⊆ ''V''. Intuitively, continuity means no matter how "small" ''V'' becomes, there is always a ''U'' containing ''x'' that maps inside ''V''. If ''X'' and ''Y'' are metric spaces, it is equivalent to consider the neighborhood system of
open ball In mathematics, a ball is the solid figure bounded by a ''sphere''; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them). These concepts are defin ...
s centered at ''x'' and ''f''(''x'') instead of all neighborhoods. This gives back the above δ-ε definition of continuity in the context of metric spaces. However, in general topological spaces, there is no notion of nearness or distance. Note, however, that if the target space is Hausdorff, it is still true that ''f'' is continuous at ''a'' if and only if the limit of ''f'' as ''x'' approaches ''a'' is ''f''(''a''). At an
isolated point ] In mathematics, a point ''x'' is called an isolated point of a subset ''S'' (in a topological space ''X'') if ''x'' is an element of ''S'' and there exists a neighborhood of ''x'' which does not contain any other points of ''S''. This is equiva ...
, every function is continuous.


Sequences and nets

In several contexts, the topology of a space is conveniently specified in terms of limit points. In many instances, this is accomplished by specifying when a point is the
limit of a sequence As the positive integer n becomes larger and larger, the value n\cdot \sin\left(\tfrac1\right) becomes arbitrarily close to 1. We say that "the limit of the sequence n\cdot \sin\left(\tfrac1\right) equals 1." In mathematics, the limit ...
, but for some spaces that are too large in some sense, one specifies also when a point is the limit of more general sets of points indexed by a
directed set In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive and transitive binary relation \,\leq\, (that is, a preorder), with the additional property that every pair of elements has ...
, known as nets. A function is continuous only if it takes limits of sequences to limits of sequences. In the former case, preservation of limits is also sufficient; in the latter, a function may preserve all limits of sequences yet still fail to be continuous, and preservation of nets is a necessary and sufficient condition. In detail, a function ''f'': ''X'' → ''Y'' is sequentially continuous if whenever a sequence (''x''''n'') in ''X'' converges to a limit ''x'', the sequence (''f''(''x''''n'')) converges to ''f''(''x''). Thus sequentially continuous functions "preserve sequential limits". Every continuous function is sequentially continuous. If ''X'' is a
first-countable space In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X is said to be first-countable if each point has a countable neighbourhood basis (local base ...
and countable choice holds, then the converse also holds: any function preserving sequential limits is continuous. In particular, if ''X'' is a metric space, sequential continuity and continuity are equivalent. For non first-countable spaces, sequential continuity might be strictly weaker than continuity. (The spaces for which the two properties are equivalent are called
sequential space In topology and related fields of mathematics, a sequential space is a topological space whose topology can be completely characterized by its convergent/divergent sequences. They can be thought of as spaces that satisfy a very weak axiom of count ...
s.) This motivates the consideration of nets instead of sequences in general topological spaces. Continuous functions preserve limits of nets, and in fact this property characterizes continuous functions.


Closure operator definition

Instead of specifying the open subsets of a topological space, the topology can also be determined by a
closure operator In mathematics, a closure operator on a set ''S'' is a function \operatorname: \mathcal(S)\rightarrow \mathcal(S) from the power set of ''S'' to itself that satisfies the following conditions for all sets X,Y\subseteq S : Closure operators are det ...
(denoted cl), which assigns to any subset ''A'' ⊆ ''X'' its closure, or an interior operator (denoted int), which assigns to any subset ''A'' of ''X'' its
interior Interior may refer to: Arts and media * ''Interior'' (Degas) (also known as ''The Rape''), painting by Edgar Degas * ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck * ''The Interior'' (novel), by Lisa See * Interior de ...
. In these terms, a function :f\colon (X,\mathrm) \to (X' ,\mathrm')\, between topological spaces is continuous in the sense above if and only if for all subsets ''A'' of ''X'' :f(\mathrm(A)) \subseteq \mathrm'(f(A)). That is to say, given any element ''x'' of ''X'' that is in the closure of any subset ''A'', ''f''(''x'') belongs to the closure of ''f''(''A''). This is equivalent to the requirement that for all subsets ''A''' of ''X''' :f^(\mathrm'(A')) \supseteq \mathrm(f^(A')). Moreover, :f\colon (X,\mathrm) \to (X' ,\mathrm') \, is continuous if and only if :f^(\mathrm'(A)) \subseteq \mathrm(f^(A)) for any subset ''A'' of ''X''.


Properties

If ''f'': ''X'' → ''Y'' and ''g'': ''Y'' → ''Z'' are continuous, then so is the composition ''g'' ∘ ''f'': ''X'' → ''Z''. If ''f'': ''X'' → ''Y'' is continuous and * ''X'' is
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Briti ...
, then ''f''(''X'') is compact. * ''X'' is connected, then ''f''(''X'') is connected. * ''X'' is
path-connected In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties t ...
, then ''f''(''X'') is path-connected. * ''X'' is Lindelöf, then ''f''(''X'') is Lindelöf. * ''X'' is separable, then ''f''(''X'') is separable. The possible topologies on a fixed set ''X'' are
partially ordered In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...
: a topology τ1 is said to be coarser than another topology τ2 (notation: τ1 ⊆ τ2) if every open subset with respect to τ1 is also open with respect to τ2. Then, the
identity map Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unch ...
:idX: (''X'', τ2) → (''X'', τ1) is continuous if and only if τ1 ⊆ τ2 (see also comparison of topologies). More generally, a continuous function :(X, \tau_X) \rightarrow (Y, \tau_Y) stays continuous if the topology τ''Y'' is replaced by a coarser topology and/or τ''X'' is replaced by a
finer topology In topology and related areas of mathematics, the set of all possible topologies on a given set forms a partially ordered set. This order relation can be used for comparison of the topologies. Definition A topology on a set may be defined as the ...
.


Homeomorphisms

Symmetric to the concept of a continuous map is an
open map In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets. That is, a function f : X \to Y is open if for any open set U in X, the image f(U) is open in Y. Likewise, a ...
, for which ''images'' of open sets are open. In fact, if an open map ''f'' has an
inverse function In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ . For a function f\colon ...
, that inverse is continuous, and if a continuous map ''g'' has an inverse, that inverse is open. Given a bijective function ''f'' between two topological spaces, the inverse function ''f''−1 need not be continuous. A bijective continuous function with continuous inverse function is called a ''
homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorph ...
''. If a continuous bijection has as its
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function ** Natural domain of a partial function ** Domain of holomorphy of a function * ...
a
compact space In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", ...
and its
codomain In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation . The term range is sometimes ambiguously used to refer to either the ...
is Hausdorff, then it is a homeomorphism.


Defining topologies via continuous functions

Given a function :f\colon X \rightarrow S, \, where ''X'' is a topological space and ''S'' is a set (without a specified topology), the
final topology In general topology and related areas of mathematics, the final topology (or coinduced, strong, colimit, or inductive topology) on a set X, with respect to a family of functions from topological spaces into X, is the finest topology on X that m ...
on ''S'' is defined by letting the open sets of ''S'' be those subsets ''A'' of ''S'' for which ''f''−1(''A'') is open in ''X''. If ''S'' has an existing topology, ''f'' is continuous with respect to this topology if and only if the existing topology is coarser than the final topology on ''S''. Thus the final topology can be characterized as the finest topology on ''S'' that makes ''f'' continuous. If ''f'' is
surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of ...
, this topology is canonically identified with the
quotient topology In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient t ...
under the
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relatio ...
defined by ''f''. Dually, for a function ''f'' from a set ''S'' to a topological space, the
initial topology In general topology and related areas of mathematics, the initial topology (or induced topology or weak topology or limit topology or projective topology) on a set X, with respect to a family of functions on X, is the coarsest topology on ''X'' t ...
on ''S'' has as open subsets ''A'' of ''S'' those subsets for which ''f''(''A'') is open in ''X''. If ''S'' has an existing topology, ''f'' is continuous with respect to this topology if and only if the existing topology is finer than the initial topology on ''S''. Thus the initial topology can be characterized as the coarsest topology on ''S'' that makes ''f'' continuous. If ''f'' is injective, this topology is canonically identified with the
subspace topology In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced t ...
of ''S'', viewed as a subset of ''X''. A topology on a set ''S'' is uniquely determined by the class of all continuous functions S \rightarrow X into all topological spaces ''X''. Dually, a similar idea can be applied to maps X \rightarrow S.


Compact sets

Formally, a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
''X'' is called ''compact'' if each of its
open cover In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alpha\ ...
s has a finite subcover. Otherwise it is called ''non-compact''. Explicitly, this means that for every arbitrary collection :\_ of open subsets of such that :X = \bigcup_ U_\alpha, there is a finite subset of such that :X = \bigcup_ U_i. Some branches of mathematics such as
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, typically influenced by the French school of Bourbaki, use the term ''quasi-compact'' for the general notion, and reserve the term ''compact'' for topological spaces that are both Hausdorff and ''quasi-compact''. A compact set is sometimes referred to as a ''compactum'', plural ''compacta''. Every closed interval in R of finite length is
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Briti ...
. More is true: In Rn, a set is compact
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...
it is
closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
and bounded. (See Heine–Borel theorem). Every continuous image of a compact space is compact. A compact subset of a Hausdorff space is closed. Every continuous
bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
from a compact space to a Hausdorff space is necessarily a
homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorph ...
. Every
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is call ...
of points in a compact metric space has a convergent subsequence. Every compact finite-dimensional
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...
can be embedded in some Euclidean space Rn.


Connected sets

A
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
''X'' is said to be disconnected if it is the union of two disjoint nonempty
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are ...
s. Otherwise, ''X'' is said to be connected. A
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
of a topological space is said to be connected if it is connected under its
subspace topology In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced t ...
. Some authors exclude the
empty set In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
(with its unique topology) as a connected space, but this article does not follow that practice. For a topological space ''X'' the following conditions are equivalent: #''X'' is connected. #''X'' cannot be divided into two disjoint nonempty
closed set In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a ...
s. #The only subsets of ''X'' that are both open and closed (
clopen set In topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open and closed. That this is possible may seem counter-intuitive, as the common meanings of and are antonyms, but their mathematical d ...
s) are ''X'' and the empty set. #The only subsets of ''X'' with empty boundary are ''X'' and the empty set. #''X'' cannot be written as the union of two nonempty
separated sets In topology and related branches of mathematics, separated sets are pairs of subsets of a given topological space that are related to each other in a certain way: roughly speaking, neither overlapping nor touching. The notion of when two sets a ...
. #The only continuous functions from ''X'' to , the two-point space endowed with the discrete topology, are constant. Every interval in R is connected. The continuous image of a connected space is connected.


Connected components

The maximal connected subsets (ordered by inclusion) of a nonempty topological space are called the connected components of the space. The components of any topological space ''X'' form a partition of ''X'': they are disjoint, nonempty, and their union is the whole space. Every component is a
closed subset In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a ...
of the original space. It follows that, in the case where their number is finite, each component is also an open subset. However, if their number is infinite, this might not be the case; for instance, the connected components of the set of the
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ratio ...
s are the one-point sets, which are not open. Let \Gamma_x be the connected component of ''x'' in a topological space ''X'', and \Gamma_x' be the intersection of all open-closed sets containing ''x'' (called quasi-component of ''x''.) Then \Gamma_x \subset \Gamma'_x where the equality holds if ''X'' is compact Hausdorff or locally connected.


Disconnected spaces

A space in which all components are one-point sets is called
totally disconnected In topology and related branches of mathematics, a totally disconnected space is a topological space that has only singletons as connected subsets. In every topological space, the singletons (and, when it is considered connected, the empty set ...
. Related to this property, a space ''X'' is called totally separated if, for any two distinct elements ''x'' and ''y'' of ''X'', there exist disjoint
open neighborhood In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. It is closely related to the concepts of open set and interior. Intuitively speaking, a neighbourhood of a po ...
s ''U'' of ''x'' and ''V'' of ''y'' such that ''X'' is the union of ''U'' and ''V''. Clearly any totally separated space is totally disconnected, but the converse does not hold. For example, take two copies of the rational numbers Q, and identify them at every point except zero. The resulting space, with the quotient topology, is totally disconnected. However, by considering the two copies of zero, one sees that the space is not totally separated. In fact, it is not even Hausdorff, and the condition of being totally separated is strictly stronger than the condition of being Hausdorff.


Path-connected sets

A '' path'' from a point ''x'' to a point ''y'' in a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
''X'' is a
continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...
''f'' from the
unit interval In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysis ...
,1to ''X'' with ''f''(0) = ''x'' and ''f''(1) = ''y''. A '' path-component'' of ''X'' is an
equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
of ''X'' under the
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relatio ...
, which makes ''x'' equivalent to ''y'' if there is a path from ''x'' to ''y''. The space ''X'' is said to be ''
path-connected In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties t ...
'' (or ''pathwise connected'' or ''0-connected'') if there is at most one path-component; that is, if there is a path joining any two points in ''X''. Again, many authors exclude the empty space. Every path-connected space is connected. The converse is not always true: examples of connected spaces that are not path-connected include the extended
long line Long line or longline may refer to: *'' Long Line'', an album by Peter Wolf *Long line (topology), or Alexandroff line, a topological space * Long line (telecommunications), a transmission line in a long-distance communications network *Longline fi ...
''L''* and the ''
topologist's sine curve In the branch of mathematics known as topology, the topologist's sine curve or Warsaw sine curve is a topological space with several interesting properties that make it an important textbook example. It can be defined as the graph of the functi ...
''. However, subsets of the
real line In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a poin ...
R are connected
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...
they are path-connected; these subsets are the
intervals Interval may refer to: Mathematics and physics * Interval (mathematics), a range of numbers ** Partially ordered set#Intervals, its generalization from numbers to arbitrary partially ordered sets * A statistical level of measurement * Interval est ...
of R. Also,
open subset In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
s of R''n'' or C''n'' are connected if and only if they are path-connected. Additionally, connectedness and path-connectedness are the same for
finite topological space In mathematics, a finite topological space is a topological space for which the underlying point set is finite. That is, it is a topological space which has only finitely many elements. Finite topological spaces are often used to provide examples ...
s.


Products of spaces

Given ''X'' such that :X := \prod_ X_i, is the Cartesian product of the topological spaces ''Xi'', indexed by i \in I, and the canonical projections ''pi'' : ''X'' → ''Xi'', the product topology on ''X'' is defined as the
coarsest topology In topology and related areas of mathematics, the set of all possible topologies on a given set forms a partially ordered set. This order relation can be used for comparison of the topologies. Definition A topology on a set may be defined as the ...
(i.e. the topology with the fewest open sets) for which all the projections ''pi'' are
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
. The product topology is sometimes called the Tychonoff topology. The open sets in the product topology are unions (finite or infinite) of sets of the form \prod_ U_i, where each ''Ui'' is open in ''Xi'' and ''U''''i'' ≠ ''X''''i'' only finitely many times. In particular, for a finite product (in particular, for the product of two topological spaces), the products of base elements of the ''Xi'' gives a basis for the product \prod_ X_i. The product topology on ''X'' is the topology generated by sets of the form ''pi''−1(''U''), where ''i'' is in ''I '' and ''U'' is an open subset of ''Xi''. In other words, the sets form a
subbase In topology, a subbase (or subbasis, prebase, prebasis) for a topological space X with topology T is a subcollection B of T that generates T, in the sense that T is the smallest topology containing B. A slightly different definition is used by so ...
for the topology on ''X''. A
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
of ''X'' is open if and only if it is a (possibly infinite) union of intersections of finitely many sets of the form ''pi''−1(''U''). The ''pi''−1(''U'') are sometimes called
open cylinder In mathematics, the cylinder sets form a basis of the product topology on a product of sets; they are also a generating family of the cylinder σ-algebra. General definition Given a collection S of sets, consider the Cartesian product X = \prod_ ...
s, and their intersections are cylinder sets. In general, the product of the topologies of each ''Xi'' forms a basis for what is called the box topology on ''X''. In general, the box topology is finer than the product topology, but for finite products they coincide. Related to compactness is
Tychonoff's theorem In mathematics, Tychonoff's theorem states that the product of any collection of compact topological spaces is compact with respect to the product topology. The theorem is named after Andrey Nikolayevich Tikhonov (whose surname sometimes is trans ...
: the (arbitrary)
product Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Produ ...
of compact spaces is compact.


Separation axioms

Many of these names have alternative meanings in some of mathematical literature, as explained on
History of the separation axioms The history of the separation axioms in general topology has been convoluted, with many meanings competing for the same terms and many terms competing for the same concept. Origins Before the current general definition of topological space, t ...
; for example, the meanings of "normal" and "T4" are sometimes interchanged, similarly "regular" and "T3", etc. Many of the concepts also have several names; however, the one listed first is always least likely to be ambiguous. Most of these axioms have alternative definitions with the same meaning; the definitions given here fall into a consistent pattern that relates the various notions of separation defined in the previous section. Other possible definitions can be found in the individual articles. In all of the following definitions, ''X'' is again a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
. * ''X'' is '' T0'', or ''Kolmogorov'', if any two distinct points in ''X'' are
topologically distinguishable In topology, two points of a topological space ''X'' are topologically indistinguishable if they have exactly the same neighborhoods. That is, if ''x'' and ''y'' are points in ''X'', and ''Nx'' is the set of all neighborhoods that contain ''x'', ...
. (It is a common theme among the separation axioms to have one version of an axiom that requires T0 and one version that doesn't.) * ''X'' is '' T1'', or ''accessible'' or ''Fréchet'', if any two distinct points in ''X'' are separated. Thus, ''X'' is T1 if and only if it is both T0 and R0. (Though you may say such things as ''T1 space'', ''Fréchet topology'', and ''Suppose that the topological space ''X'' is Fréchet'', avoid saying ''Fréchet space'' in this context, since there is another entirely different notion of
Fréchet space In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces (normed vector spaces that are complete with respect to t ...
in
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined ...
.) * ''X'' is '' Hausdorff'', or ''T2'' or ''separated'', if any two distinct points in ''X'' are separated by neighbourhoods. Thus, ''X'' is Hausdorff if and only if it is both T0 and R1. A Hausdorff space must also be T1. * ''X'' is '' T2½'', or ''Urysohn'', if any two distinct points in ''X'' are separated by closed neighbourhoods. A T2½ space must also be Hausdorff. * ''X'' is '' regular'', or ''T3'', if it is T0 and if given any point ''x'' and closed set ''F'' in ''X'' such that ''x'' does not belong to ''F'', they are separated by neighbourhoods. (In fact, in a regular space, any such ''x'' and ''F'' is also separated by closed neighbourhoods.) * ''X'' is '' Tychonoff'', or ''T3½'', ''completely T3'', or ''completely regular'', if it is T0 and if f, given any point ''x'' and closed set ''F'' in ''X'' such that ''x'' does not belong to ''F'', they are separated by a continuous function. * ''X'' is '' normal'', or ''T4'', if it is Hausdorff and if any two disjoint closed subsets of ''X'' are separated by neighbourhoods. (In fact, a space is normal if and only if any two disjoint closed sets can be separated by a continuous function; this is
Urysohn's lemma In topology, Urysohn's lemma is a lemma that states that a topological space is normal if and only if any two disjoint closed subsets can be separated by a continuous function. Section 15. Urysohn's lemma is commonly used to construct continu ...
.) * ''X'' is ''
completely normal In topology and related branches of mathematics, a normal space is a topological space ''X'' that satisfies Axiom T4: every two disjoint closed sets of ''X'' have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space. T ...
'', or ''T5'' or ''completely T4'', if it is T1 and if any two separated sets are separated by neighbourhoods. A completely normal space must also be normal. * ''X'' is '' perfectly normal'', or ''T6'' or ''perfectly T4'', if it is T1 and if any two disjoint closed sets are precisely separated by a continuous function. A perfectly normal Hausdorff space must also be completely normal Hausdorff. The
Tietze extension theorem In topology, the Tietze extension theorem (also known as the Tietze–Urysohn–Brouwer extension theorem) states that continuous functions on a closed subset of a normal topological space can be extended to the entire space, preserving boundednes ...
: In a normal space, every continuous real-valued function defined on a closed subspace can be extended to a continuous map defined on the whole space.


Countability axioms

An axiom of countability is a
property Property is a system of rights that gives people legal control of valuable things, and also refers to the valuable things themselves. Depending on the nature of the property, an owner of property may have the right to consume, alter, share, r ...
of certain
mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deductive reasoning and mathematical p ...
s (usually in a
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) *Category (Kant) *Categories (Peirce) * ...
) that requires the existence of a
countable set In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers ...
with certain properties, while without it such sets might not exist. Important countability axioms for
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
s: *
sequential space In topology and related fields of mathematics, a sequential space is a topological space whose topology can be completely characterized by its convergent/divergent sequences. They can be thought of as spaces that satisfy a very weak axiom of count ...
: a set is open if every
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is call ...
convergent to a point in the set is eventually in the set *
first-countable space In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X is said to be first-countable if each point has a countable neighbourhood basis (local base ...
: every point has a countable
neighbourhood basis In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter \mathcal(x) for a point x in a topological space is the collection of all neighbourhoods of x. Definitions Neighbour ...
(local base) * second-countable space: the topology has a countable base *
separable space In mathematics, a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence \_^ of elements of the space such that every nonempty open subset of the space contains at least one element of t ...
: there exists a countable dense subspace *
Lindelöf space In mathematics, a Lindelöf space is a topological space in which every open cover has a countable subcover. The Lindelöf property is a weakening of the more commonly used notion of ''compactness'', which requires the existence of a ''finite'' sub ...
: every
open cover In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alpha\ ...
has a countable subcover *
σ-compact space In mathematics, a topological space is said to be σ-compact if it is the union of countably many compact subspaces. A space is said to be σ-locally compact if it is both σ-compact and locally compact. Properties and examples * Every compact ...
: there exists a countable cover by compact spaces Relations: *Every first countable space is sequential. *Every second-countable space is first-countable, separable, and Lindelöf. *Every σ-compact space is Lindelöf. *A
metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
is first-countable. *For metric spaces second-countability, separability, and the Lindelöf property are all equivalent.


Metric spaces

A metric space is an
ordered pair In mathematics, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In con ...
(M,d) where M is a set and d is a
metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathema ...
on M, i.e., a function :d \colon M \times M \rightarrow \mathbb such that for any x, y, z \in M, the following holds: # d(x,y) \ge 0     (''non-negative''), # d(x,y) = 0\, iff x = y\,     (''
identity of indiscernibles The identity of indiscernibles is an ontological principle that states that there cannot be separate objects or entities that have all their properties in common. That is, entities ''x'' and ''y'' are identical if every predicate possessed by ''x ...
''), # d(x,y) = d(y,x)\,     (''symmetry'') and # d(x,z) \le d(x,y) + d(y,z)     (''
triangle inequality In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of degenerate triangles, but ...
'') . The function d is also called ''distance function'' or simply ''distance''. Often, d is omitted and one just writes M for a metric space if it is clear from the context what metric is used. Every
metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
is
paracompact In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is norma ...
and Hausdorff, and thus normal. The metrization theorems provide necessary and sufficient conditions for a topology to come from a metric.


Baire category theorem

The
Baire category theorem The Baire category theorem (BCT) is an important result in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space (a topological space such that t ...
says: If ''X'' is a complete metric space or a
locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ...
Hausdorff space, then the interior of every union of
countably many In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...
nowhere dense In mathematics, a subset of a topological space is called nowhere dense or rare if its closure has empty interior. In a very loose sense, it is a set whose elements are not tightly clustered (as defined by the topology on the space) anywhere. ...
sets is empty. Any open subspace of a
Baire space In mathematics, a topological space X is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior. According to the Baire category theorem, compact Hausdorff spaces and complete metric spaces are e ...
is itself a Baire space.


Main areas of research


Continuum theory

A continuum (pl ''continua'') is a nonempty
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Briti ...
connected
metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
, or less frequently, a
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Briti ...
connected Hausdorff space. Continuum theory is the branch of topology devoted to the study of continua. These objects arise frequently in nearly all areas of topology and
analysis Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (384 ...
, and their properties are strong enough to yield many 'geometric' features.


Dynamical systems

Topological dynamics concerns the behavior of a space and its subspaces over time when subjected to continuous change. Many examples with applications to physics and other areas of math include
fluid dynamics In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) an ...
,
billiards Cue sports are a wide variety of games of skill played with a cue, which is used to strike billiard balls and thereby cause them to move around a cloth-covered table bounded by elastic bumpers known as . There are three major subdivisions ...
and flows on manifolds. The topological characteristics of
fractal In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as il ...
s in fractal geometry, of
Julia set In the context of complex dynamics, a branch of mathematics, the Julia set and the Fatou set are two complementary sets (Julia "laces" and Fatou "dusts") defined from a function. Informally, the Fatou set of the function consists of values wit ...
s and the
Mandelbrot set The Mandelbrot set () is the set of complex numbers c for which the function f_c(z)=z^2+c does not diverge to infinity when iterated from z=0, i.e., for which the sequence f_c(0), f_c(f_c(0)), etc., remains bounded in absolute value. Thi ...
arising in
complex dynamics Complex dynamics is the study of dynamical systems defined by iteration of functions on complex number spaces. Complex analytic dynamics is the study of the dynamics of specifically analytic functions. Techniques *General ** Montel's theorem ** ...
, and of
attractor In the mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain ...
s in differential equations are often critical to understanding these systems.


Pointless topology

Pointless topology (also called point-free or pointfree topology) is an approach to
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
that avoids mentioning points. The name 'pointless topology' is due to
John von Neumann John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest cove ...
.Garrett Birkhoff, ''VON NEUMANN AND LATTICE THEORY'', ''John Von Neumann 1903-1957'', J. C. Oxtoley, B. J. Pettis, American Mathematical Soc., 1958, page 50-5 The ideas of pointless topology are closely related to mereotopologies, in which regions (sets) are treated as foundational without explicit reference to underlying point sets.


Dimension theory

Dimension theory is a branch of general topology dealing with dimensional invariants of
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
s.


Topological algebras

A topological algebra ''A'' over a
topological field In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is ...
K is a
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
together with a continuous multiplication :\cdot :A\times A \longrightarrow A :(a,b)\longmapsto a\cdot b that makes it an
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...
over K. A unital
associative In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacemen ...
topological algebra is a
topological ring In mathematics, a topological ring is a ring R that is also a topological space such that both the addition and the multiplication are continuous as maps: R \times R \to R where R \times R carries the product topology. That means R is an additive ...
. The term was coined by
David van Dantzig David van Dantzig (September 23, 1900 – July 22, 1959) was a Dutch mathematician, well known for the construction in topology of the dyadic solenoid. He was a member of the Significs Group. Biography Born to a Jewish family in Amsterdam in ...
; it appears in the title of his doctoral dissertation (1931).


Metrizability theory

In
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
and related areas of mathematics, a metrizable space is a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
that is
homeomorphic In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorp ...
to a
metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
. That is, a topological space (X,\tau) is said to be metrizable if there is a metric :d\colon X \times X \to ,\infty) such that the topology induced by ''d'' is \tau. Metrization theorems are theorems that give sufficient conditions for a topological space to be metrizable.


Set-theoretic topology

Set-theoretic topology is a subject that combines set theory and general topology. It focuses on topological questions that are independent of
Zermelo–Fraenkel set theory In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such a ...
(ZFC). A famous problem is the normal Moore space question, a question in general topology that was the subject of intense research. The answer to the normal Moore space question was eventually proved to be independent of ZFC.


See also

*
List of examples in general topology {{Short description, none This is a list of useful examples in general topology, a field of mathematics. * Alexandrov topology * Cantor space * Co-kappa topology ** Cocountable topology ** Cofinite topology * Compact-open topology * Compactifica ...
* Glossary of general topology for detailed definitions *
List of general topology topics This is a list of general topology topics, by Wikipedia page. Basic concepts *Topological space * Topological property *Open set, closed set **Clopen set **Closure (topology) **Boundary (topology) **Dense (topology) ** G-delta set, F-sigma set * ...
for related articles *
Category of topological spaces In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again conti ...


References


Further reading

Some standard books on general topology include: * Bourbaki, Topologie Générale (General Topology), . *
John L. Kelley John L. Kelley (December 6, 1916, Kansas – November 26, 1999, Berkeley, California) was an American mathematician at the University of California, Berkeley, who worked in general topology and functional analysis. Kelley's 1955 text, ''General ...
(1955
''General Topology''
link from
Internet Archive The Internet Archive is an American digital library with the stated mission of "universal access to all knowledge". It provides free public access to collections of digitized materials, including websites, software applications/games, music, ...
, originally published by
David Van Nostrand David Van Nostrand (December 5, 1811 – June 14, 1886) was a New York City publisher. Biography David Van Nostrand was born in New York City on December 5, 1811. He was educated at Union Hall, Jamaica, New York, and in 1826 entered the publis ...
Company. * Stephen Willard, General Topology, . *
James Munkres James Raymond Munkres (born August 18, 1930) is a Professor Emeritus of mathematics at MIT and the author of several texts in the area of topology, including ''Topology'' (an undergraduate-level text), ''Analysis on Manifolds'', ''Elements of Alge ...
, Topology, . * George F. Simmons, Introduction to Topology and Modern Analysis, . * Paul L. Shick, Topology: Point-Set and Geometric, . *
Ryszard Engelking Ryszard Engelking (born 1935-11-16 in Sosnowiec) is a Polish mathematician. He was working mainly on general topology and dimension theory. He is author of several influential monographs in this field. The 1989 edition of his ''General Topology'' ...
, General Topology, . * * O.Ya. Viro, O.A. Ivanov, V.M. Kharlamov and N.Yu. Netsvetaev
Elementary Topology: Textbook in Problems
. The
arXiv arXiv (pronounced "archive"—the X represents the Greek letter chi ⟨χ⟩) is an open-access repository of electronic preprints and postprints (known as e-prints) approved for posting after moderation, but not peer review. It consists of ...
subject code i
math.GN


External links

* {{Areas of mathematics , collapsed