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In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including
differential topology In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which ...
,
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 ...
. 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", ...
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 ...
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 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 spaces, in the early days of functional analysis. 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 of subsets 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 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 ** ''U ...
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 ''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 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-class ...
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 ''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 which the open sets are the intersections of the open sets of the larger space with the subset. For any indexed family of topological spaces, the product can be given the product topology, 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 on ''Y'' is the collection of subsets of ''Y'' that have open inverse images 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 to ...
, 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 (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 many ...
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 cocou ...
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 numbers. 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 spaces R''n'' can be given a topology. In the usual topology on R''n'' the basic open sets are the open
ball A ball is a round object (usually spherical, but can sometimes be ovoid) with several uses. It is used in ball games, where the play of the game follows the state of the ball as it is hit, kicked or thrown by players. Balls can also be used fo ...
s. Similarly, C, the set of complex numbers, 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. 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 example ...
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, since it is locally Euclidean. Similarly, every simplex and every simplicial complex inherits a natural topology from Rn. * The Zariski topology 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 set In mathematics, a solution set is the set of values that satisfy a given set of equations or inequalities. For example, for a set of polynomials over a ring , the solution set is the subset of on which the polynomials all vanish (evaluate t ...
s of systems of polynomial equations. * A
linear graph In the mathematical field of graph theory, a path graph or linear graph is a graph whose vertices can be listed in the order such that the edges are where . Equivalently, a path with at least two vertices is connected and has two terminal ...
has a natural topology that generalises many of the geometric aspects of graphs with vertices and edges. * Many sets of
linear operator In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
s in functional analysis are endowed with topologies that are defined by specifying when a particular sequence of functions converges to the zero function. * Any local field 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

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_Continuous_functions

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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

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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 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 Neighbou ...
of open balls 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, every function is continuous.


Sequences and nets

In several contexts, the topology of a space is conveniently specified in terms of
limit points In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also conta ...
. 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, 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 and
countable choice The axiom of countable choice or axiom of denumerable choice, denoted ACω, is an axiom of set theory that states that every countable collection of non-empty sets must have a choice function. That is, given a function ''A'' with domain N (where ...
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 spaces.) 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 d ...
(denoted cl), which assigns to any subset ''A'' ⊆ ''X'' its closure, or an
interior 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 d ...
(denoted int), which assigns to any subset ''A'' of ''X'' its interior. 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 Britis ...
, 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: 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, unc ...
:idX: (''X'', τ2) → (''X'', τ1) is continuous if and only if τ1 ⊆ τ2 (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 :(X, \tau_X) \rightarrow (Y, \tau_Y) 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 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 ...
.


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 ...
, 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 isomorph ...
''. 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 ...
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 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 under the equivalence relation defined by ''f''. Dually, for a function ''f'' from a set ''S'' to a topological space, the initial topology 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 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_\alp ...
s has a
finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb Traditionally, a finite verb (from la, fīnītus, past particip ...
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 geometrica ...
, 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 Britis ...
. More is true: In Rn, a set is compact if and only if 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 ...
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 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 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 ** ''U ...
of two disjoint
nonempty 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 ...
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 of a topological space is said to be connected if it is connected under its subspace topology. 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 de ...
s) are ''X'' and the empty set. #The only subsets of ''X'' with empty
boundary Boundary or Boundaries may refer to: * Border, in political geography Entertainment * ''Boundaries'' (2016 film), a 2016 Canadian film * ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film * Boundary (cricket), the edge of the pl ...
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 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 ...
. The continuous image of a
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 ...
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 numbers 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. 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 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 A path is a route for physical travel – see Trail. Path or PATH may also refer to: Physical paths of different types * Bicycle path * Bridle path, used by people on horseback * Course (navigation), the intended path of a vehicle * Desire p ...
'' 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 ''f'' from the unit interval ,1to ''X'' with ''f''(0) = ''x'' and ''f''(1) = ''y''. A '' 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'' (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''. 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 they are path-connected; these subsets are the 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 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 example ...
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 th ...
(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 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 ** ''U ...
of
intersections 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 sets of the form ''pi''−1(''U''). The ''pi''−1(''U'') are sometimes called open cylinders, and their intersections are
cylinder set 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. In general, the product of the topologies of each ''Xi'' forms a basis for what is called the
box topology In topology, the cartesian product of topological spaces can be given several different topologies. One of the more obvious choices is the box topology, where a base is given by the Cartesian products of open sets in the component spaces. Another p ...
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: 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; 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. (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.) * ''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 '' T'', or ''Urysohn'', if any two distinct points in ''X'' are separated by closed neighbourhoods. A T 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 ''T'', ''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.) * ''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 ''Perfectly Normal'' is a Canadian comedy film directed by Yves Simoneau, which premiered at the 1990 Festival of Festivals, before going into general theatrical release in 1991. Simoneau's first English-language film, it was written by Eugene Li ...
'', 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 of certain mathematical objects (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) *C ...
) 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: a set is open if every sequence convergent to a point in the set is eventually in the set * first-countable space: every point has a countable neighbourhood basis (local base) *
second-countable space In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space T is second-countable if there exists some countable collection \mat ...
: the topology has a countable base * separable space: there exists a countable dense subspace * Lindelöf space: 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_\alp ...
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 (M,d) where M is a set and d is a metric on M, i.e., a
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 ...
: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 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 bicond ...
x = y\,     ('' identity of indiscernibles''), # d(x,y) = d(y,x)\,     (''symmetry'') and # d(x,z) \le d(x,y) + d(y,z)     ('' triangle inequality'') . 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 and Hausdorff, and thus normal. The
metrization theorems In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) ...
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 the ...
says: If ''X'' is a
complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
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 nowhere dense 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 ...
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 Britis ...
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 ...
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 Britis ...
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 ...
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 many ...
. 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 (3 ...
, 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,
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 o ...
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 sets 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 **P ...
, 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 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 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 :\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 a ...
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 replacement ...
topological algebra is a topological ring. The term was coined by David van Dantzig; it appears in the title of his
doctoral dissertation A thesis ( : theses), or dissertation (abbreviated diss.), is a document submitted in support of candidature for an academic degree or professional qualification presenting the author's research and findings.International Standard ISO 7144: ...
(1931).


Metrizability theory

In topology 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 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 continu ...


References


Further reading

Some standard books on general topology include: * Bourbaki, Topologie Générale (General Topology), . * John L. Kelley (1955
''General Topology''
link from Internet Archive, originally published by David Van Nostrand Company. *
Stephen Willard Stephen Willard (born 27 August 1958 in Swindon) is a former professional English darts player. Who played in Professional Darts Corporation events. He won a PDC Tour Card in 2015, which was the year he also won the Saints Open defeating G ...
, 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 Alg ...
, Topology, . *
George F. Simmons George Finlay Simmons (March 3, 1925 – August 6, 2019) was an American mathematician who worked in topology and classical analysis. He is known as the author of widely used textbooks on university mathematics. Life He was born on 3 March 192 ...
, Introduction to Topology and Modern Analysis, . *
Paul L. Shick Paul may refer to: * Paul (given name), a given name (includes a list of people with that name) *Paul (surname), a list of people People Christianity *Paul the Apostle (AD c.5–c.64/65), also known as Saul of Tarsus or Saint Paul, early Chri ...
, 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 s ...
subject code i
math.GN


External links

* {{Areas of mathematics , collapsed