View source for General topology

{{Short description|Branch of topology}}
[[Image:Topologist's sine curve.svg|thumb|420px|The [[Topologist's sine curve]], a useful example in point-set topology. It is connected but not path-connected.]]

In [[mathematics]], '''general topology''' (or '''point set topology''') is the branch of [[topology]] that deals with the basic [[Set theory|set-theoretic]] definitions and constructions used in topology. It is the foundation of most other branches of topology, including [[differential topology]], [[geometric topology]], and [[algebraic topology]].

The fundamental concepts in point-set topology are ''continuity'', ''compactness'', and ''connectedness'': 
* [[Continuous function]]s, intuitively, take nearby points to nearby points. 
* [[Compact set]]s are those that can be covered by finitely many sets of arbitrarily small size.
* [[Connected set]]s are sets that cannot be divided into two pieces that are far apart. 
The terms 'nearby', 'arbitrarily small', and 'far apart' can all be made precise by using the concept of [[open set]]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]]''.

''[[Metric space]]s'' are an important class of topological spaces where a real, non-negative distance, also called a ''[[metric (mathematics)|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]]s, especially [[normed linear space]]s, 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 [[continuous function|continuity]], in a technically adequate form that can be applied in any area of mathematics.

==A topology on a set==
{{Main|Topological space}}
Let ''X'' be a set and let ''τ'' be a [[Family of sets|family]] of [[subset]]s of ''X''. Then ''τ'' is called a ''topology on X'' if:<ref>Munkres, James R. Topology. Vol. 2. Upper Saddle River: Prentice Hall, 2000.</ref><ref>Adams, Colin Conrad, and Robert David Franzosa. Introduction to topology: pure and applied. Pearson Prentice Hall, 2008.</ref>

# Both the [[empty set]] and ''X'' are elements of ''τ''
# Any [[union (set theory)|union]] of elements of ''τ'' is an element of ''τ''
# Any [[intersection (set theory)|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<sub>τ</sub>'' may be used to denote a set ''X'' endowed with the particular topology ''τ''.

The members of ''τ'' are called ''[[open set]]s'' in ''X''. A subset of ''X'' is said to be [[closed set|closed]] if its [[Complement (set theory)|complement]] is in ''τ'' (i.e., its complement is open). A subset of ''X'' may be open, closed, both ([[clopen set]]), or neither. The empty set and ''X'' itself are always both closed and open.

===Basis for a topology===
{{Main|Basis (topology)}}
A '''base''' (or '''basis''') ''B'' for a [[topological space]] ''X'' with [[topological space|topology]] ''T'' is a collection of [[open set]]s in ''T'' such that every open set in ''T'' can be written as a union of elements of ''B''.<ref>{{cite book |last1=Merrifield |first1=Richard E. |last2=Simmons |first2=Howard E. |author-link2=Howard Ensign Simmons Jr. |title=Topological Methods in Chemistry |year=1989 |publisher=John Wiley & Sons |location=New York |isbn=0-471-83817-9 |url=https://archive.org/details/topologicalmetho00merr/page/16 |access-date=27 July 2012 |pages=[https://archive.org/details/topologicalmetho00merr/page/16 16] |quote='''Definition.''' A collection ''B'' of subsets of a topological space ''(X,T)'' is called a ''basis'' for ''T'' if every open set can be expressed as a union of members of ''B''. |url-access=registration }}</ref><ref>{{cite book |last=Armstrong |first=M. A. |title=Basic Topology |year=1983 |publisher=Springer |isbn=0-387-90839-0  |url=https://www.springer.com/mathematics/geometry/book/978-0-387-90839-7 |access-date=13 June 2013 |page=30 |quote=Suppose we have a topology on a set ''X'', and a collection <math>\beta</math> of open sets such that every open set is a union of members of <math>\beta</math>. Then <math>\beta</math> is called a ''base'' for the topology...}}</ref>  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 (mathematics)|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 (topology)|quotient space]] is defined as follows: if ''X'' is a topological space and ''Y'' is a set, and if ''f'' : ''X''→ ''Y'' is a [[surjection|surjective]] [[function (mathematics)|function]], then the [[quotient topology]] on ''Y'' is the collection of subsets of ''Y'' that have open [[inverse image]]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 class]]es.

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

====Discrete and trivial topologies====
Any set can be given the [[discrete space|discrete topology]], in which every subset is open.  The only convergent sequences or nets in this topology are those that are eventually constant.  Also, any set can be given the [[trivial topology]] (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]]s where limit points are unique.

====Cofinite and cocountable topologies====
Any set can be given the [[cofinite topology]] in which the open sets are the empty set and the sets whose complement is finite.  This is the smallest [[T1 space|T<sub>1</sub>]] topology on any infinite set.

Any set can be given the [[cocountable topology]], in which a set is defined as open if it is either empty or its complement is countable.  When the set is uncountable, this topology serves as a counterexample in many situations.

====Topologies on the real and complex numbers====
There are many ways to define a topology on '''R''', the set of [[real number]]s.  The standard topology on '''R''' is generated by the [[Interval (mathematics)#Definitions|open intervals]].  The set of all open intervals forms a [[base (topology)|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]]s '''R'''<sup>''n''</sup> can be given a topology.  In the usual topology on '''R'''<sup>''n''</sup> the basic open sets are the open [[Ball (mathematics)|ball]]s.  Similarly, '''C''', the set of [[complex number]]s, and '''C'''<sup>''n''</sup> 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 <nowiki>[</nowiki>''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]] 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]]s. Finite spaces are sometimes used to provide examples or counterexamples to conjectures about topological spaces in general.
* Every [[manifold]] has a [[natural topology]], since it is locally Euclidean.  Similarly, every [[simplex]] and every [[simplicial complex]] inherits a natural topology from '''R'''<sup>n</sup>.
* The [[Zariski topology]] is defined algebraically on the [[spectrum of a ring]] or an [[algebraic variety]].  On '''R'''<sup>''n''</sup> or '''C'''<sup>''n''</sup>, the closed sets of the Zariski topology are the [[solution set]]s of systems of [[polynomial]] equations.
* A [[linear graph]] has a natural topology that generalises many of the geometric aspects of [[graph theory|graph]]s with [[Vertex (graph theory)|vertices]] and [[Graph (discrete mathematics)#Graph|edges]].
* Many sets of [[linear operator]]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]] is the simplest non-discrete topological space.  It has important relations to the [[theory of computation]] and semantics.
* If Γ is an [[ordinal number]], then the set Γ = [0,&nbsp;Γ) may be endowed with the [[order topology]] generated by the intervals (''a'',&nbsp;''b''), [0,&nbsp;''b'') and (''a'',&nbsp;Γ) where ''a'' and ''b'' are elements of Γ.

==Continuous functions==
{{Main|Continuous function}}

Continuity is expressed in terms of [[neighborhood (topology)|neighborhood]]s: {{math|''f''}} is continuous at some point {{math|''x''&nbsp;∈&nbsp;''X''}} if and only if for any neighborhood {{math|''V''}} of {{math|''f''(''x'')}}, there is a neighborhood {{math|''U''}} of {{math|''x''}} such that {{math|''f''(''U'')&nbsp;⊆&nbsp;''V''}}. Intuitively, continuity means no matter how "small" {{math|''V''}} becomes, there is always a {{math|''U''}} containing {{math|''x''}} that maps inside {{math|''V''}} and whose image under {{math|''f''}} contains {{math|''f''(''x'')}}. This is equivalent to the condition that the [[Image (mathematics)#Inverse image|preimages]] of the open (closed) sets in {{math|''Y''}} are open (closed) in {{math|''X''}}. In metric spaces, this definition is equivalent to the [[epsilon-delta definition|ε&ndash;δ-definition]] that is often used in analysis.

An extreme example: if a set {{math|''X''}} is given the [[discrete topology]], all functions 
:<math>f\colon X \rightarrow T</math>
to any topological space {{math|''T''}} are continuous. On the other hand, if {{math|''X''}} is equipped with the [[indiscrete topology]] and the space {{math|''T''}} set is at least [[T0 space|T<sub>0</sub>]], 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 (topology)|neighborhood]]s: ''f'' is continuous at some point ''x''&nbsp;∈&nbsp;''X'' if and only if for any neighborhood ''V'' of ''f''(''x''), there is a neighborhood ''U'' of ''x'' such that ''f''(''U'')&nbsp;⊆&nbsp;''V''. Intuitively, continuity means no matter how "small" ''V'' becomes, there is always a ''U'' containing ''x'' that maps inside ''V''.

If ''X'' and ''Y'' are metric spaces, it is equivalent to consider the [[neighborhood system]] of [[open ball]]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 space|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 {{anchor|Heine definition of continuity}}====
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]], known as [[net (mathematics)|nets]].<ref>{{Cite journal | doi = 10.2307/2370388 | last1 = Moore | first1 = E. H. | last2 = Smith | first2 = H. L. | author1-link = E. H. Moore | author2-link = Herman L. Smith | year = 1922 | title = A General Theory of Limits | journal = American Journal of Mathematics | volume = 44 | issue = 2 | pages = 102&ndash;121 | jstor = 2370388}}</ref>  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''<sub>''n''</sub>) in ''X'' converges to a limit ''x'', the sequence (''f''(''x''<sub>''n''</sub>)) converges to ''f''(''x'').<ref>{{cite journal | last1 = Heine | first1 = E. | year = 1872 | title = Die Elemente der Functionenlehre.. | url = http://eudml.org/doc/148175 | journal = Journal für die reine und angewandte Mathematik | volume = 74 | pages = 172–188 }}</ref>  Thus sequentially continuous functions "preserve sequential limits".  Every continuous function is sequentially continuous.  If ''X'' is a [[first-countable space]] and [[Axiom of countable choice|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]]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 [[Kuratowski closure operator|closure operator]] (denoted cl), which assigns to any subset ''A'' ⊆ ''X'' its [[closure (topology)|closure]], or an [[interior operator]] (denoted int), which assigns to any subset ''A'' of ''X'' its [[interior (topology)|interior]]. In these terms, a function 
:<math>f\colon (X,\mathrm{cl}) \to (X' ,\mathrm{cl}')\, </math>
between topological spaces is continuous in the sense above if and only if for all subsets ''A'' of ''X''
:<math>f(\mathrm{cl}(A)) \subseteq \mathrm{cl}'(f(A)).</math>
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''<nowiki>'</nowiki> of ''X''<nowiki>'</nowiki>
:<math>f^{-1}(\mathrm{cl}'(A')) \supseteq \mathrm{cl}(f^{-1}(A')).</math>
Moreover, 
:<math>f\colon (X,\mathrm{int}) \to (X' ,\mathrm{int}') \, </math>
is continuous if and only if 
:<math>f^{-1}(\mathrm{int}'(A)) \subseteq \mathrm{int}(f^{-1}(A))</math>
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 space|compact]], then ''f''(''X'') is compact.
* ''X'' is [[Connected space|connected]], then ''f''(''X'') is connected.
* ''X'' is [[path-connected]], then ''f''(''X'') is path-connected.
* ''X'' is [[Lindelöf space|Lindelöf]], then ''f''(''X'') is Lindelöf.
* ''X'' is [[separable space|separable]], then ''f''(''X'') is separable.

The possible topologies on a fixed set ''X'' are [[partial ordering|partially ordered]]: a topology τ<sub>1</sub> is said to be [[comparison of topologies|coarser]] than another topology τ<sub>2</sub> (notation: τ<sub>1</sub> ⊆ τ<sub>2</sub>) if every open subset with respect to τ<sub>1</sub> is also open with respect to τ<sub>2</sub>. Then, the [[identity function|identity map]] 
:id<sub>X</sub>: (''X'', τ<sub>2</sub>) → (''X'', τ<sub>1</sub>)
is continuous if and only if τ<sub>1</sub> ⊆ τ<sub>2</sub> (see also [[comparison of topologies]]). More generally, a continuous function
:<math>(X, \tau_X) \rightarrow (Y, \tau_Y)</math>
stays continuous if the topology τ<sub>''Y''</sub> is replaced by a [[Comparison of topologies|coarser topology]] and/or τ<sub>''X''</sub> is replaced by a [[Comparison of topologies|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]], that inverse is continuous, and if a continuous map ''g'' has an inverse, that inverse is open. Given a [[bijective]] function ''f'' between two topological spaces, the inverse function ''f''<sup>&minus;1</sup> need not be continuous. A bijective continuous function with continuous inverse function is called a ''[[homeomorphism]]''.

If a continuous bijection has as its [[Domain of a function|domain]] a [[compact space]] and its [[codomain]] is [[Hausdorff space|Hausdorff]], then it is a homeomorphism.

===Defining topologies via continuous functions===
Given a function
:<math>f\colon X \rightarrow S, \,</math>
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''<sup>−1</sup>(''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 [[Comparison of topologies|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 ''X'', the [[initial topology]] on ''S'' has a basis of open sets given by those sets of the form ''f^(-1)''(''U'') where ''U''  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 <math>S \rightarrow X</math> into all topological spaces ''X''. [[Duality (mathematics)|Dually]], a similar idea can be applied to maps <math>X \rightarrow S.</math>

==Compact sets==
{{Main|Compact (mathematics)}}
Formally, a [[topological space]] ''X'' is called ''compact'' if each of its [[open cover]]s has a [[finite set|finite]] [[subcover]]. Otherwise it is called ''non-compact''.  Explicitly, this means that for every arbitrary collection

:<math>\{U_\alpha\}_{\alpha\in A}</math>

of open subsets of {{mvar|X}} such that

:<math>X = \bigcup_{\alpha\in A} U_\alpha,</math>

there is a finite subset {{mvar|J}} of {{mvar|A}} such that

:<math>X = \bigcup_{i\in J} U_i.</math>

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'''<sup><var>n</var></sup>, a set is compact [[if and only if]] it is [[closed set|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]].

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'''<sup><var>n</var></sup>.

==Connected sets==
{{Main|connected space}}
A [[topological space]] ''X'' is said to be '''disconnected''' if it is the [[union (set theory)|union]] of two [[disjoint sets|disjoint]] [[nonempty]] [[open set]]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 set]]s.
#The only subsets of ''X'' that are both open and closed ([[clopen set]]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 {0,1}, 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&nbsp;''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 number]]s are the one-point sets, which are not open.

Let <math>\Gamma_x</math> be the connected component of ''x'' in a topological space ''X'', and <math>\Gamma_x'</math> be the intersection of all open-closed sets containing ''x'' (called [[Locally connected space|quasi-component]] of ''x''.) Then <math>\Gamma_x \subset \Gamma'_x</math> 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 neighborhood]]s ''U'' of ''x'' and ''V'' of ''y'' such that ''X'' is the union of ''U'' and ''V''. Clearly any totally separated space is totally disconnected, but the converse does not hold. For example, take two copies of the rational numbers '''Q''', and identify them at every point except zero. The resulting space, with the quotient topology, is totally disconnected. However, by considering the two copies of zero, one sees that the space is not totally separated.  In fact, it is not even [[Hausdorff space|Hausdorff]], and the condition of being totally separated is strictly stronger than the condition of being Hausdorff.

===Path-connected sets===
[[File:Path-connected space.svg|thumb|This subspace of '''R'''² is path-connected, because a path can be drawn between any two points in the space.]]

A ''[[path (topology)|path]]'' from a point ''x'' to a point ''y''  in a [[topological space]] ''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 subset]]s of '''R'''<sup>''n''</sup> or '''C'''<sup>''n''</sup> are connected if and only if they are path-connected. Additionally, connectedness and path-connectedness are the same for [[finite topological space]]s.

==Products of spaces==
{{Main|Product topology}}
Given ''X'' such that

:<math>X := \prod_{i \in I} X_i,</math>

is the Cartesian product of the topological spaces ''X<sub>i</sub>'', [[index set|indexed]] by <math>i \in I</math>, and the '''[[projection (set theory)|canonical projections]]''' ''p<sub>i</sub>'' : ''X'' &rarr; ''X<sub>i</sub>'', 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<sub>i</sub>'' 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 <math>\prod_{i\in I} U_i</math>, where each ''U<sub>i</sub>'' is open in ''X<sub>i</sub>'' and ''U''<sub>''i''</sub>&nbsp;≠&nbsp;''X''<sub>''i''</sub> 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<sub>i</sub>'' gives a basis for the product <math>\prod_{i\in I} X_i</math>.

The product topology on ''X'' is the topology generated by sets of the form ''p<sub>i</sub>''<sup>&minus;1</sup>(''U''), where  ''i'' is in ''I ''  and ''U'' is an open subset of ''X<sub>i</sub>''. In other words, the sets {''p<sub>i</sub>''<sup>&minus;1</sup>(''U'')} form a [[subbase]] for the topology on ''X''. A [[subset]] of ''X'' is open if and only if it is a (possibly infinite) [[union (set theory)|union]] of [[intersection (set theory)|intersections]] of finitely many sets of the form ''p<sub>i</sub>''<sup>&minus;1</sup>(''U''). The ''p<sub>i</sub>''<sup>&minus;1</sup>(''U'') are sometimes called [[open cylinder]]s, and their intersections are [[cylinder set]]s.

In general, the product of the topologies of each ''X<sub>i</sub>'' 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==
{{Main|Separation axiom}}
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<sub>4</sub>" are sometimes interchanged, similarly "regular" and "T<sub>3</sub>", 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]].

* ''X'' is ''[[T0 space|T<sub>0</sub>]]'', 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<sub>0</sub> and one version that doesn't.)
* ''X'' is ''[[T1 space|T<sub>1</sub>]]'', or ''accessible'' or ''Fréchet'', if any two distinct points in ''X'' are separated. Thus, ''X'' is T<sub>1</sub> if and only if it is both T<sub>0</sub> and R<sub>0</sub>. (Though you may say such things as ''T<sub>1</sub> 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 space|Hausdorff]]'', or ''T<sub>2</sub>'' 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<sub>0</sub> and R<sub>1</sub>. A Hausdorff space must also be T<sub>1</sub>.
* ''X'' is ''[[Urysohn and completely Hausdorff spaces|T<sub>2½</sub>]]'', or ''Urysohn'', if any two distinct points in ''X'' are separated by closed neighbourhoods. A T<sub>2½</sub> space must also be Hausdorff.
* ''X'' is ''[[regular space|regular]]'', or ''T<sub>3</sub>'', if it is T<sub>0</sub> 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<sub>3½</sub>'', ''completely T<sub>3</sub>'', or ''completely regular'', if it is T<sub>0</sub> 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<sub>4</sub>'', 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<sub>5</sub>'' or ''completely T<sub>4</sub>'', if it is T<sub>1</sub> 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<sub>6</sub>'' or ''perfectly T<sub>4</sub>'', if it is T<sub>1</sub> 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==
{{Main|axiom of countability}}
An '''axiom of countability''' is a [[property]] of certain [[mathematical object]]s (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]]s:
*[[sequential space]]: 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]]: every point has a countable [[neighbourhood system|neighbourhood basis]] (local  base)
*[[second-countable space]]: the topology has a countable [[base (topology)|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]] is first-countable.
*For metric spaces second-countability, separability, and the Lindelöf property are all equivalent.

==Metric spaces==
{{Main|Metric space}}
A '''metric space'''<ref>[[Maurice Fréchet]] introduced metric spaces in his work ''Sur quelques points du calcul fonctionnel'', Rendic. Circ. Mat. Palermo 22 (1906) 1–74.</ref> is an [[ordered pair]] <math>(M,d)</math> where <math>M</math> is a set and <math>d</math> is a [[metric (mathematics)|metric]] on <math>M</math>, i.e., a [[Function (mathematics)|function]]

:<math>d \colon M \times M \rightarrow \mathbb{R}</math>

such that for any <math>x, y, z \in M</math>, the following holds:

# <math>d(x,y) \ge 0</math> &nbsp;&nbsp;&nbsp; (''non-negative''), 
# <math>d(x,y) = 0\,</math> [[if and only if|iff]] <math>x = y\,</math> &nbsp;&nbsp;&nbsp; (''[[identity of indiscernibles]]''),
# <math>d(x,y) = d(y,x)\,</math> &nbsp;&nbsp;&nbsp; (''symmetry'') and
# <math>d(x,z) \le d(x,y) + d(y,z)</math> &nbsp;&nbsp;&nbsp; (''[[triangle inequality]]'') .

The function <math>d</math> is also called ''distance function'' or simply ''distance''. Often, <math>d</math> is omitted and one just writes <math>M</math> for a metric space if it is clear from the context what metric is used.

Every [[metric space]] is [[paracompact]] and [[Hausdorff space|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==
{{Main|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.<ref>R. Baire. [https://books.google.com/books?id=cS4LAAAAYAAJ Sur les fonctions de variables réelles.] Ann. di Mat., 3:1–123, 1899.</ref>

Any open subspace of a [[Baire space]] is itself a Baire space.

==Main areas of research==
[[Image:Peanocurve.svg|400px|thumb|Three iterations of a Peano curve construction, whose limit is a space-filling curve. The Peano curve is studied in [[continuum theory]], a branch of '''general topology'''.]]

===Continuum theory===
{{Main|Continuum (topology)}}
A '''continuum''' (pl ''continua'') is a nonempty [[compact space|compact]] [[connected space|connected]] [[metric space]], or less frequently, a [[compact space|compact]] [[connected space|connected]] [[Hausdorff space]]. '''Continuum theory''' is the branch of topology devoted to the study of continua.  These objects arise frequently in nearly all areas of topology and [[mathematical analysis|analysis]], and their properties are strong enough to yield many 'geometric' features.

===Dynamical systems===
{{Main|Topological dynamics}}
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 [[fractal]]s in fractal geometry, of [[Julia set]]s and the [[Mandelbrot set]] arising in [[complex dynamics]], and of [[attractor]]s in differential equations are often critical to understanding these systems.{{Citation needed|date=December 2019}}

===Pointless topology===
{{Main|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]].<ref>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</ref>  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===
{{Main|Dimension theory}}
'''Dimension theory''' is a branch of general topology dealing with [[dimensional invariant]]s of [[topological space]]s.

===Topological algebras===
{{Main|Topological algebra}}
A '''topological algebra''' ''A'' over a [[topological field]] '''K''' is a [[topological vector space]] together with a continuous multiplication

:<math>\cdot :A\times A \longrightarrow A</math>
:<math>(a,b)\longmapsto a\cdot b</math>

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===
{{Main|Metrization theorem}}
In [[topology]] and related areas of [[mathematics]], a '''metrizable space''' is a [[topological space]] that is [[homeomorphism|homeomorphic]] to a [[metric space]]. That is, a topological space <math>(X,\tau)</math> is said to be metrizable if there is a metric 
:<math>d\colon X \times X \to [0,\infty)</math>

such that the topology induced by ''d'' is <math>\tau</math>.  '''Metrization theorems''' are [[theorem]]s that give [[sufficient condition]]s for a topological space to be metrizable.

===Set-theoretic topology===
{{Main|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 ==
{{reflist}}

== Further reading ==
Some standard books on general topology include:
* [[Nicolas Bourbaki|Bourbaki]],          <cite>Topologie Générale</cite> (<cite>General Topology</cite>), {{ISBN|0-387-19374-X}}.
* {{Kelley 1975}}
* Stephen Willard,   <cite>General Topology</cite>, {{ISBN|0-486-43479-6}}.
* [[James Munkres]],     <cite>Topology</cite>, {{ISBN|0-13-181629-2}}.
* [[George F. Simmons]], <cite>Introduction to Topology and Modern Analysis</cite>, {{ISBN|1-575-24238-9}}.
* [[Paul L. Shick]],     <cite>Topology: Point-Set and Geometric</cite>, {{ISBN|0-470-09605-5}}.
* [[Ryszard Engelking]], <cite>General Topology</cite>, {{ISBN|3-88538-006-4}}.
* {{citation |last1=Steen |first1=Lynn Arthur |author1-link=Lynn Arthur Steen |last2=Seebach |first2=J. Arthur Jr. |author2-link=J. Arthur Seebach Jr. |title=Counterexamples in Topology |title-link=Counterexamples in Topology |orig-year=1978 |publisher=[[Springer-Verlag]] |location=Berlin, New York |edition=[[Dover Publications|Dover]] reprint of 1978 |isbn=978-0-486-68735-3 |mr=507446 |year=1995}}
* O.Ya. Viro, O.A. Ivanov, V.M. Kharlamov and N.Yu. Netsvetaev, [https://www.ams.org/bookstore-getitem/item=mbk-54 <cite>Elementary Topology: Textbook in Problems</cite>], {{ISBN|978-0-8218-4506-6}}.

The [[arXiv]] subject code is [https://arxiv.org/list/math.GN/recent math.GN].

==External links==
*{{Commons category-inline}}

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{{Areas of mathematics |collapsed}}
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