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 concern ...
definitions and constructions used in topology. It is the foundation of most other branches of topology, including
differential topology,
geometric topology, 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 ...
. Another name for general topology is point-set topology.
The fundamental concepts in point-set topology are ''continuity'', ''compactness'', and ''connectedness'':
*
Continuous functions, 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", i. ...
s are those that can be covered by finitely many sets of arbitrarily small size.
*
Connected sets 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 su ...
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'', 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 (once known as the ''topology of point sets''; this usage is now obsolete)
*the introduction of the
manifold 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 o ...
.
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. Idea ...
of
subsets of ''X''. Then ''τ'' is called a ''topology on X'' if:
# Both the
empty set and ''X'' are elements of ''τ''
# Any
union
Union commonly refers to:
* Trade union, an organization of workers
* Union (set theory), in mathematics, a fundamental operation on sets
Union may also refer to:
Arts and entertainment
Music
* Union (band), an American rock group
** ''Un ...
of elements of ''τ'' is an element of ''τ''
# Any
intersection 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 ''X
τ'' 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 su ...
s'' in ''X''. A subset of ''X'' is said to be
closed if its
complement
A complement is something that completes something else.
Complement may refer specifically to:
The arts
* Complement (music), an interval that, when added to another, spans an octave
** Aggregate complementation, the separation of pitch-clas ...
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 de ...
), 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 su ...
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 to ...
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, wher ...
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-s ...
, 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 function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
, 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 to ...
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 is defined on the topological space ''X''. The map ''f'' is then the natural projection onto the set of
equivalence classes.
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 topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are ''isolated'' from each other in a certain sense. The discrete topology is the finest top ...
, 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 conseque ...
(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 space
In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the m ...
s where limit points are unique.
Cofinite and cocountable topologies
Any set can be given the
cofinite topology
In mathematics, a cofinite subset of a set X is a subset A whose complement in X is a finite set. In other words, A contains all but finitely many elements of X. If the complement is not finite, but it is countable, then one says the set is cocoun ...
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 The cocountable topology or countable complement topology on any set ''X'' consists of the empty set and all cocountable subsets of ''X'', that is all sets whose complement in ''X'' is countable. It follows that the only closed subsets are ''X'' and ...
, 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. Every ...
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 fo ...
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 spaces. Finite spaces are sometimes used to provide examples or counterexamples to conjectures about topological spaces in general.
* Every
manifold 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 and every
simplicial complex inherits a natural topology from R
n.
* 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 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. ...
. 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 example ...
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 discre ...
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 o ...
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 name ...
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 algorithmic efficiency, efficiently they can be solved or t ...
and semantics.
* If Γ is an
ordinal number, then the set Γ =
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_Continuous_functions
Continuity_is_expressed_in_terms_of_
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_Continuous_functions
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_Continuous_functions
Continuity_is_expressed_in_terms_of_neighborhood_(topology)">neighborhoods:__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)">neighborhoods: 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
:
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 space, T
0, 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
neighborhoods: ''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 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 ...
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 defi ...
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, 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 ha ...
, 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 counta ...
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 de ...
(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. In these terms, a function
:
between topological spaces is continuous in the sense above if and only if for all subsets ''A'' of ''X''
:
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''
'
:
Moreover,
:
is continuous if and only if
:
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, then ''f''(''X'') is compact.
* ''X'' is
connected
Connected may refer to:
Film and television
* ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular''
* '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film
* ''Connected'' (2015 TV ...
, then ''f''(''X'') is connected.
* ''X'' is
path-connected, 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 r ...
: a topology τ
1 is said to be
coarser than another topology τ
2 (notation: τ
1 ⊆ τ
2) if every open subset with respect to τ
1 is also open with respect to τ
2. Then, the
identity map
:id
X: (''X'', τ
2) → (''X'', τ
1)
is continuous if and only if τ
1 ⊆ τ
2 (see also
comparison of topologies
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 th ...
). More generally, a continuous function
:
stays continuous if the topology τ
''Y'' is replaced by a
coarser 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 th ...
and/or τ
''X'' is replaced by a
finer topology.
Homeomorphisms
Symmetric to the concept of a continuous map is an
open map, 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 X ...
, that inverse is continuous, and if a continuous map ''g'' has an inverse, that inverse is open. Given a
bijective
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 ...
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 isomor ...
''.
If a continuous bijection has as its
domain 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 th ...
is
Hausdorff, then it is a homeomorphism.
Defining topologies via continuous functions
Given a function
:
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 make ...
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, 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 to ...
under the
equivalence relation 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 to ...
of ''S'', viewed as a subset of ''X''.
A topology on a set ''S'' is uniquely determined by the class of all continuous functions
into all topological spaces ''X''. Duality (mathematics), Dually, a similar idea can be applied to maps
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 covers has a finite set, finite subcover. Otherwise it is called ''non-compact''. Explicitly, this means that for every arbitrary collection
:
of open subsets of such that
:
there is a finite subset of such that
:
Some branches of mathematics such as algebraic geometry, typically influenced by the French school of Nicolas Bourbaki, Bourbaki, use the term ''quasi-compact'' for the general notion, and reserve the term ''compact'' for topological spaces that are both Hausdorff spaces, Hausdorff and ''quasi-compact''. A compact set is sometimes referred to as a ''compactum'', plural ''compacta''.
Every closed interval (mathematics), interval in Real number, R of finite length is compact space, compact. More is true: In R
n, a set is compact if and only if it is
closed 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 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 isomor ...
.
Every sequence of points in a compact metric space has a convergent subsequence.
Every compact finite-dimensional
manifold can be embedded in some Euclidean space R
n.
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
Union commonly refers to:
* Trade union, an organization of workers
* Union (set theory), in mathematics, a fundamental operation on sets
Union may also refer to:
Arts and entertainment
Music
* Union (band), an American rock group
** ''Un ...
of two disjoint sets, 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 su ...
s. Otherwise, ''X'' is said to be connected. A
subset of a topological space is said to be connected if it is connected under its subspace (topology), subspace topology. Some authors exclude the
empty set (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 sets.
#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 de ...
s) are ''X'' and the empty set.
#The only subsets of ''X'' with empty boundary (topology), boundary are ''X'' and the empty set.
#''X'' cannot be written as the union of two nonempty separated sets.
#The only continuous functions from ''X'' to , the two-point space endowed with the discrete topology, are constant.
Every interval in R is connected space, connected.
The continuous image of a connectedness, connected space is connected.
Connected components
The maximal element, maximal connected subsets (ordered by subset, 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 a set, partition of ''X'': they are disjoint sets, disjoint, nonempty, and their union is the whole space.
Every component is a closed subset 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 numbers are the one-point sets, which are not open.
Let
be the connected component of ''x'' in a topological space ''X'', and
be the intersection of all open-closed sets containing ''x'' (called Locally connected space, quasi-component of ''x''.) Then
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. 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 neighborhood (topology), open neighborhoods ''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 (topology), 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 (topology), continuous function ''f'' from the unit interval [0,1] to ''X'' with ''f''(0) = ''x'' and ''f''(1) = ''y''. A ''Path component, path-component'' of ''X'' is an
equivalence class of ''X'' under the
equivalence relation, which makes ''x'' equivalent to ''y'' if there is a path from ''x'' to ''y''. The space ''X'' is said to be ''path-connected space, path-connected'' (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 (topology), long line ''L''* and the ''topologist's sine curve''.
However, subsets of the
real line R are connected if and only if they are path-connected; these subsets are the interval (mathematics), intervals of R. Also, open subsets 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 spaces.
Products of spaces
Given ''X'' such that
:
is the Cartesian product of the topological spaces ''X
i'', index set, indexed by
, and the projection (set theory), canonical projections ''p
i'' : ''X'' → ''X
i'', the product topology on ''X'' is defined as the coarsest topology (i.e. the topology with the fewest open sets) for which all the projections ''p
i'' are continuous (topology), continuous. 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
, where each ''U
i'' is open in ''X
i'' 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 ''X
i'' gives a basis for the product
.
The product topology on ''X'' is the topology generated by sets of the form ''p
i''
−1(''U''), where ''i'' is in ''I '' and ''U'' is an open subset of ''X
i''. In other words, the sets form a subbase for the topology on ''X''. A
subset of ''X'' is open if and only if it is a (possibly infinite)
union
Union commonly refers to:
* Trade union, an organization of workers
* Union (set theory), in mathematics, a fundamental operation on sets
Union may also refer to:
Arts and entertainment
Music
* Union (band), an American rock group
** ''Un ...
of intersection (set theory), intersections of finitely many sets of the form ''p
i''
−1(''U''). The ''p
i''
−1(''U'') are sometimes called open cylinders, and their intersections are cylinder sets.
In general, the product of the topologies of each ''X
i'' forms a basis for what is called the box topology on ''X''. In general, the box topology is finer topology, finer than the product topology, but for finite products they coincide.
Related to compactness is Tychonoff's theorem: the (arbitrary) product topology, product 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; for example, the meanings of "normal" and "T
4" are sometimes interchanged, similarly "regular" and "T
3", 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 space, T
0'', or ''Kolmogorov'', if any two distinct points in ''X'' are topological distinguishability, topologically distinguishable. (It is a common theme among the separation axioms to have one version of an axiom that requires T
0 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 T
1 if and only if it is both T
0 and R
0. (Though you may say such things as ''T
1 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
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 o ...
.)
* ''X'' is ''
Hausdorff'', or ''T
2'' or ''separated'', if any two distinct points in ''X'' are separated by neighbourhoods. Thus, ''X'' is Hausdorff if and only if it is both T
0 and R
1. A Hausdorff space must also be T
1.
* ''X'' is ''Urysohn and completely Hausdorff spaces, T
2½'', or ''Urysohn'', if any two distinct points in ''X'' are separated by closed neighbourhoods. A T
2½ space must also be Hausdorff.
* ''X'' is ''regular space, regular'', or ''T
3'', if it is T
0 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 space, Tychonoff'', or ''T
3½'', ''completely T
3'', or ''completely regular'', if it is T
0 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 space, normal'', or ''T
4'', 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.)
* ''X'' is ''completely normal space, completely normal'', or ''T
5'' or ''completely T
4'', if it is T
1 and if any two separated sets are separated by neighbourhoods. A completely normal space must also be normal.
* ''X'' is ''perfectly normal space, perfectly normal'', or ''T
6'' or ''perfectly T
4'', if it is T
1 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 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 of certain mathematical objects (usually in a Category (mathematics), category) that requires the existence of a countable, countable set 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 counta ...
: a set is open if every sequence limit of a sequence, convergent to a point (geometry), 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 system, neighbourhood basis (local base)
*second-countable space: the topology has a countable
base
*separable space: there exists a countable dense (topology), dense subspace
*Lindelöf space: every open cover has a countable subcover
*σ-compact space: 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
where
is a set and
is a
metric on
, i.e., a Function (mathematics), function
:
such that for any
, the following holds:
#
(''non-negative''),
#
if and only if, iff
(''identity of indiscernibles''),
#
(''symmetry'') and
#
(''triangle inequality'') .
The function
is also called ''distance function'' or simply ''distance''. Often,
is omitted and one just writes
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 and
Hausdorff, and thus normal space, normal.
The metrization theorems provide necessary and sufficient conditions for a topology to come from a metric.
Baire category theorem
The Baire category theorem says: If ''X'' is a completeness (topology), complete metric space or a locally compact Hausdorff space, then the interior of every union of countable, countably many nowhere dense sets is empty.
Any open subspace of a Baire space is itself a Baire space.
Main areas of research
Continuum theory
A continuum (pl ''continua'') is a nonempty compact space, compact connected space, 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 space, compact connected space, connected
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the m ...
. Continuum theory is the branch of topology devoted to the study of continua. These objects arise frequently in nearly all areas of topology and mathematical analysis, analysis, 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, dynamical billiards, billiards and geometric flow, flows on manifolds. The topological characteristics of fractals in fractal geometry, of Julia sets and the Mandelbrot set arising in complex dynamics, and of attractors 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.
[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 mereotopology, 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 K is a topological vector space together with a continuous multiplication
:
:
that makes it an algebra over a field, algebra over K. A unital associative algebra, associative topological algebra is a topological ring.
The term was coined by David van Dantzig; it appears in the title of his Thesis, 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 homeomorphism, homeomorphic 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
is said to be metrizable if there is a metric
:
such that the topology induced by ''d'' is
. 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 (ZFC). A famous problem is Moore space (topology)#Normal Moore space conjecture, 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
*Glossary of general topology for detailed definitions
*List of general topology topics for related articles
*Category of topological spaces
References
Further reading
Some standard books on general topology include:
* Nicolas Bourbaki, Bourbaki,
Topologie Générale (
General Topology), .
* John L. Kelley (1955
''General Topology'' link from Internet Archive, originally published by David Van Nostrand Company.
* Stephen Willard,
General Topology, .
* James Munkres,
Topology, .
* George F. Simmons,
Introduction to Topology and Modern Analysis, .
* Paul L. Shick,
Topology: Point-Set and Geometric, .
* Ryszard Engelking,
General Topology, .
*
* O.Ya. Viro, O.A. Ivanov, V.M. Kharlamov and N.Yu. Netsvetaev
Elementary Topology: Textbook in Problems .
The arXiv subject code i
math.GN
External links
*
{{Areas of mathematics , collapsed
General topology,