View source for Topological space

{{Short description|Mathematical space with a notion of closeness}}
In [[mathematics]], a '''topological space''' is, roughly speaking, a [[Geometry|geometrical space]] in which [[Closeness (mathematics)|closeness]] is defined but cannot necessarily be measured by a numeric [[Distance (mathematics)|distance]]. More specifically, a topological space is a [[Set (mathematics)|set]] whose elements are called [[Point (geometry)|point]]s, along with an additional structure called a topology, which can be defined as a set of [[Neighbourhood (mathematics)|neighbourhood]]s for each point that satisfy some [[Axiom#Non-logical axioms|axiom]]s formalizing the concept of closeness. There are several equivalent definitions of a topology, the most commonly used of which is the definition through [[open set]]s.

A topological space is the most general type of a [[space (mathematics)|mathematical space]] that allows for the definition of [[Limit (mathematics)|limits]], [[Continuous function (topology)|continuity]], and [[Connected space|connectedness]].<ref>{{harvnb|Schubert|1968|loc=p. 13}}</ref><ref>{{Cite book|last=Sutherland|first=W. A.|title=Introduction to metric and topological spaces|date=1975|publisher=Clarendon Press|isbn=0-19-853155-9|location=Oxford [England]|oclc=1679102}}</ref> Common types of topological spaces include [[Euclidean space]]s, [[metric space]]s and [[manifold]]s.

Although very general, the concept of topological spaces is fundamental, and used in virtually every branch of modern mathematics. The study of topological spaces in their own right is called [[general topology]] (or point-set topology).

== History ==
Around 1735, [[Leonhard Euler]] discovered the [[Planar graph#Euler's formula|formula]] <math>V - E + F = 2</math> relating the number of vertices (V), edges (E) and faces (F) of a [[Convex polytope|convex polyhedron]], and hence of a [[planar graph]]. The study and generalization of this formula, specifically by [[Augustin-Louis Cauchy| Cauchy]] (1789–1857) and [[Simon Antoine Jean L'Huilier| L'Huilier]] (1750–1840), [[Euler's Gem | boosted the study]] of topology. In 1827, [[Carl Friedrich Gauss]] published ''General investigations of curved surfaces'', which in section 3 defines the curved surface in a similar manner to the modern topological understanding: "A curved surface is said to possess continuous curvature at one of its points A, if the direction of all the straight lines drawn from A to points of the surface at an infinitesimal distance from A are deflected infinitesimally from one and the same plane passing through A."{{sfn|Gauss|1827}}{{psi|date=June 2024}}

Yet, "until [[Bernhard Riemann| Riemann]]'s work in the early 1850s, surfaces were always dealt with from a local point of view (as parametric surfaces) and topological issues were never considered".{{sfn|Gallier|Xu|2013}} "[[August Ferdinand Möbius| Möbius]] and [[Camille Jordan| Jordan]] seem to be the first to realize that the main problem about the topology of (compact) surfaces is to find invariants (preferably numerical) to decide the equivalence of surfaces, that is, to decide whether two surfaces are [[homeomorphism | homeomorphic]] or not."{{sfn|Gallier|Xu|2013}}

The subject is clearly defined by [[Felix Klein]] in his "[[Erlangen Program]]" (1872): the geometry invariants of arbitrary continuous transformation, a kind of geometry. The term "topology" was introduced by [[Johann Benedict Listing]] in 1847, although he had used the term in correspondence some years earlier instead of previously used "Analysis situs". The foundation of this science, for a space of any dimension, was created by [[Henri Poincaré]]. His first article on this topic appeared in 1894.<ref>J. Stillwell, Mathematics and its history</ref> In the 1930s, [[James Waddell Alexander II]] and [[Hassler Whitney]] first expressed the idea that a surface is a topological space that is [[Topological manifold |locally like a Euclidean plane]].

Topological spaces were first defined by [[Felix Hausdorff]] in 1914 in his seminal "Principles of Set Theory". [[Metric spaces]] had been defined earlier in 1906 by [[Maurice Fréchet]], though it was Hausdorff who popularised the term "metric space" ({{langx |de| metrischer Raum}}).<ref>
{{oed | metric space}}
</ref><ref>
{{cite book
 |last1                = Hausdorff
 |first1               = Felix
 |author-link1         = Felix Hausdorff
 |orig-date            = 1914
 |chapter              = Punktmengen in allgemeinen Räumen
 |title                = Grundzüge der Mengenlehre
 |url                  = https://books.google.com/books?id=xMZXAAAAYAAJ
 |series               = Göschens Lehrbücherei/Gruppe I: Reine und Angewandte Mathematik Serie
 |date = 1914
 |language             = de
 |location             = Leipzig
 |publisher            = Von Veit
 |publication-date     = 2011
 |page                 = 211
 |isbn                 = 9783110989854
 |access-date          = 20 August 2022
 |quote                = Unter einem m e t r i s c h e n &nbsp; R a u m e verstehen wir eine Menge ''E'', [...].
}} 
</ref>{{better source|date=June 2024}}

== Definitions ==
{{main|Axiomatic foundations of topological spaces}}

The utility of the concept of a ''topology'' is shown by the fact that there are several equivalent definitions of this [[mathematical structure]]. Thus one chooses the [[axiomatization]] suited for the application. The most commonly used is that in terms of {{em|[[open set]]s}}, but perhaps more intuitive is that in terms of {{em|[[Neighbourhood (mathematics)|neighbourhood]]s}} and so this is given first.

=== Definition via neighbourhoods{{anchor|Neighborhood definition|Neighbourhood definition}} ===

This axiomatization is due to [[Felix Hausdorff]].
Let <math>X</math> be a (possibly empty) set. The elements of <math>X</math> are usually called {{em|points}}, though they can be any mathematical object. Let <math>\mathcal{N}</math> be a [[Function (mathematics)|function]] assigning to each <math>x</math> (point) in <math>X</math> a non-empty collection <math>\mathcal{N}(x)</math> of subsets of <math>X.</math> The elements of <math>\mathcal{N}(x)</math> will be called {{em|neighbourhoods}} of <math>x</math> with respect to <math>\mathcal{N}</math> (or, simply, {{em|neighbourhoods of <math>x</math>}}). The function <math>\mathcal{N}</math> is called a [[Neighbourhood (topology)|neighbourhood topology]] if the [[axiom]]s below{{sfn|Brown|2006|loc=section 2.1}} are satisfied; and then <math>X</math> with <math>\mathcal{N}</math> is called a '''topological space'''.

# If <math>N</math> is a neighbourhood of <math>x</math> (i.e., <math>N \in \mathcal{N}(x)</math>), then <math>x \in N.</math>  In other words, each point of the set <math>X</math> belongs to every one of its neighbourhoods with respect to <math> \mathcal{N} </math>.
# If <math>N</math> is a subset of <math>X</math> and includes a neighbourhood of <math>x,</math> then <math>N</math> is a neighbourhood of <math>x.</math>  I.e., every [[superset]] of a neighbourhood of a point <math>x \in X</math> is again a neighbourhood of <math>x.</math>
# The [[Intersection (set theory)|intersection]] of two neighbourhoods of <math>x</math> is a neighbourhood of <math>x.</math>
# Any neighbourhood <math>N</math> of <math>x</math> includes a neighbourhood <math>M</math> of <math>x</math> such that <math>N</math> is a neighbourhood of each point of <math>M.</math>

The first three axioms for neighbourhoods have a clear meaning. The fourth axiom has a very important use in the structure of the theory, that of linking together the neighbourhoods of different points of <math>X.</math>

A standard example of such a system of neighbourhoods is for the real line <math>\R,</math> where a subset <math>N</math> of <math>\R</math> is defined to be a {{em|neighbourhood}} of a real number <math>x</math> if it includes an open interval containing <math>x.</math>

Given such a structure, a subset <math>U</math> of <math>X</math> is defined to be '''open''' if <math>U</math> is a neighbourhood of all points in <math>U.</math> The open sets then satisfy the axioms given below in the next definition of a topological space. Conversely, when given the open sets of a topological space, the neighbourhoods satisfying the above axioms can be recovered by defining <math>N</math> to be a neighbourhood of <math>x</math> if <math>N</math> includes an open set <math>U</math> such that <math>x \in U.</math>{{sfn|Brown|2006|loc=section 2.2}}

=== Definition via open sets {{anchor|topology}} ===
{{anchor|topological space}}
A ''topology'' on a [[Set (mathematics)|set]] {{mvar|X}} may be defined as a collection <math>\tau</math> of [[subset]]s of {{mvar|X}}, called '''open sets''' and satisfying the following axioms:{{sfn|Armstrong|1983|loc=definition 2.1}}

# The [[empty set]] and <math>X</math> itself belong to <math>\tau.</math>
# Any arbitrary (finite or infinite) [[Union (set theory)|union]] of members of <math>\tau</math> belongs to <math>\tau.</math>
# The intersection of any finite number of members of <math>\tau</math> belongs to <math>\tau.</math>

As this definition of a topology is the most commonly used, the set <math>\tau</math> of the open sets is commonly called a '''topology''' on <math>X.</math> 

A subset <math>C \subseteq X</math> is said to be {{em|closed}} in <math>(X, \tau)</math> if its [[Complement (set theory)|complement]] <math>X \setminus C</math> is an open set. Note that from this definition, it follows that the empty set and <math>X</math> are simultaneously open ''and'' closed – that is, the two sets are complements of one another, while each of them is, itself, open. In general, any subset of <math>X</math> with this property is said to be ''[[Clopen set|clopen]].''

==== Examples of topologies ====

[[Image:Topological space examples.svg|frame|right|Let <math>\tau</math> be denoted with the circles, here are four examples and two non-examples of topologies on the three-point set <math>\{1,2,3\}.</math> The bottom-left example is not a topology because the union of <math>\{2\}</math> and <math>\{3\}</math> [i.e. <math>\{2,3\}</math>] is missing; the bottom-right example is not a topology because the intersection of <math>\{1,2\}</math> and <math>\{2,3\}</math> [i.e. <math>\{2\}</math>], is missing.]]

# Given <math>X = \{ 1, 2, 3, 4\},</math> the [[Trivial topology|trivial]] or {{em|indiscrete}} topology on <math>X</math> is the [[Family of sets|family]] <math>\tau = \{ \{ \}, \{ 1, 2, 3, 4 \} \} = \{ \varnothing, X \}</math> consisting of only the two subsets of <math>X</math> required by the axioms forms a topology on <math>X.</math>
# Given <math>X = \{ 1, 2, 3, 4\},</math> the family <math display="block">\tau = \{ \varnothing, \{ 2 \}, \{1, 2\}, \{2, 3\}, \{1, 2, 3\}, X \}</math> of six subsets of <math>X</math> forms another topology of <math>X.</math>
# Given <math>X = \{ 1, 2, 3, 4\},</math> the [[discrete topology]] on <math>X</math> is the [[power set]] of <math>X,</math> which is the family <math>\tau = \wp(X)</math> consisting of all possible subsets of <math>X.</math> In this case the topological space <math>(X, \tau)</math> is called a ''[[discrete space]]''.
# Given <math>X = \Z,</math> the set of integers, the family <math>\tau</math> of all finite subsets of the integers plus <math>\Z</math> itself is {{em|not}} a topology, because (for example) the union of all finite sets not containing zero is not finite and therefore not a member of the family of finite sets. The union of all finite sets not containing zero is also not all of <math>\Z,</math> and so it cannot be in <math>\tau.</math>

=== Definition via closed sets ===
Using [[de Morgan's laws]], the above axioms defining open sets become axioms defining '''[[closed set]]s''':

# The empty set and <math>X</math> are closed.
# The intersection of any collection of closed sets is also closed.
# The union of any finite number of closed sets is also closed.

Using these axioms, another way to define a topological space is as a set <math>X</math> together with a collection <math>\tau</math> of closed subsets of <math>X.</math> Thus the sets in the topology <math>\tau</math> are the closed sets, and their complements in <math>X</math> are the open sets.

=== Other definitions ===
There are many other equivalent ways to define a topological space: in other words the concepts of neighbourhood, or that of open or closed sets can be reconstructed from other starting points and satisfy the correct axioms.

Another way to define a topological space is by using the [[Kuratowski closure axioms]], which define the closed sets as the [[Fixed point (mathematics)|fixed points]] of an [[Operator (mathematics)|operator]] on the [[power set]] of <math>X.</math>

A [[Net (mathematics)|net]] is a generalisation of the concept of [[sequence]].  A topology is completely determined if for every net in <math>X</math> the set of its [[Topology glossary|accumulation points]] is specified.

== Comparison of topologies ==
{{main|Comparison of topologies}}
{{citations needed|section|date=May 2025}}
Many topologies can be defined on a set to form a topological space. When every open set of a topology <math>\tau_1</math> is also open for a topology <math>\tau_2,</math> one says that <math>\tau_2</math> is {{em|finer}} than <math>\tau_1,</math> and <math>\tau_1</math> is {{em|coarser}} than <math>\tau_2.</math> A proof that relies only on the existence of certain open sets will also hold for any finer topology, and similarly a proof that relies only on certain sets not being open applies to any coarser topology.  The terms {{em|larger}} and {{em|smaller}} are sometimes used in place of finer and coarser, respectively.  The terms {{em|stronger}} and {{em|weaker}} are also used in the literature, but with little agreement on the meaning, so one should always be sure of an author's convention when reading.

The collection of all topologies on a given fixed set <math>X</math> forms a [[complete lattice]]: if <math>F = \left\{ \tau_{\alpha} : \alpha \in A \right\}</math> is a collection of topologies on <math>X,</math> then the [[Join and meet|meet]] of <math>F</math> is the intersection of <math>F,</math> and the [[Join and meet|join]] of <math>F</math> is the meet of the collection of all topologies on <math>X</math> that contain every member of <math>F.</math>

== Continuous functions ==
{{main|Continuous function}}

A [[Function (mathematics)|function]] <math>f : X \to Y</math> between topological spaces is called '''[[Continuity (topology)|continuous]]''' if for every <math> x \in X</math> and every neighbourhood <math>N</math> of <math>f(x)</math> there is a neighbourhood <math>M</math> of <math>x</math> such that <math>f(M) \subseteq N.</math> This relates easily to the usual definition in analysis. Equivalently, <math>f</math> is continuous if the [[inverse image]] of every open set is open.{{sfn|Armstrong|1983|loc=theorem 2.6}}  This is an attempt to capture the intuition that there are no "jumps" or "separations" in the function.  A [[homeomorphism]] is a [[bijection]] that is continuous and whose [[Inverse function|inverse]] is also continuous.  Two spaces are called {{em|homeomorphic}} if there exists a homeomorphism between them.  From the standpoint of topology, homeomorphic spaces are essentially identical.<ref>{{Cite book|isbn = 978-93-325-4953-1|last = Munkres|first = James R|title = Topology|date = 2015|pages = 317–319| publisher=Pearson }}</ref>

In [[category theory]], one of the fundamental [[Category (mathematics)|categories]] is '''Top''', which denotes the [[category of topological spaces]] whose [[Object (category theory)|objects]] are topological spaces and whose [[morphism]]s are continuous functions.  The attempt to classify the objects of this category ([[up to]] [[homeomorphism]]) by [[Invariant (mathematics)|invariant]]s has motivated areas of research, such as [[Homotopy|homotopy theory]], [[Homology (mathematics)|homology theory]], and [[K-theory]].{{cn|date=May 2025}}

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

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.

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.<ref>{{cite journal
 | last1 = Anderson | first1 = B. A.
 | last2 = Stewart | first2 = D. G.
 | doi = 10.2307/2037491
 | journal = Proceedings of the American Mathematical Society
 | jstor = 2037491
 | mr = 244927
 | pages = 77–81
 | title = <math>T_1</math>-complements of <math>T_1</math> topologies
 | volume = 23
 | year = 1969}}</ref>

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.

The real line can also be given the [[lower limit topology]].  Here, the basic open sets are the half open intervals <math>[a, b).</math>  This topology on <math>\R</math> 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.

If <math>\gamma</math> is an [[ordinal number]], then the set <math>\gamma = [0, \gamma)</math> may be endowed with the [[order topology]] generated by the intervals <math>(\alpha, \beta),</math> <math>[0, \beta),</math> and <math>(\alpha, \gamma)</math> where <math>\alpha</math> and <math>\beta</math> are elements of <math>\gamma.</math>

Every [[manifold]] has a [[natural topology]] since it is locally Euclidean.  Similarly, every [[simplex]] and every [[simplicial complex]] inherits a natural topology from .

The [[Sierpiński space]] is the simplest non-discrete topological space.  It has important relations to the [[theory of computation]] and semantics.

=== Topology from other topologies ===
{{citations needed|section|date=May 2025}}
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. This construction is a special case of an [[initial topology]].

A [[Quotient space (topology)|quotient space]] is defined as follows: if <math>X</math> is a topological space and <math>Y</math> is a set, and if <math>f : X \to Y</math> is a [[Surjection|surjective]] [[Function (mathematics)|function]], then the quotient topology on <math>Y</math> is the collection of subsets of <math>Y</math> that have open [[inverse image]]s under <math>f.</math> In other words, the quotient topology is the finest topology on <math>Y</math> for which <math>f</math> is continuous.  A common example of a quotient topology is when an [[equivalence relation]] is defined on the topological space <math>X.</math>  The map <math>f</math> is then the natural projection onto the set of [[equivalence class]]es. This construction is a special case of a [[final topology]].

=== Metric spaces ===
{{main|Metric space}}

Metric spaces embody a [[Metric (mathematics)|metric]], a precise notion of distance between points.

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.

There are many ways of defining a topology on <math>\R,</math> the set of [[real number]]s.  The standard topology on <math>\R</math> 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 <math>\R^n</math> can be given a topology.  In the '''usual topology''' on <math>\R^n</math> the basic open sets are the open [[Ball (mathematics)|ball]]s.  Similarly, <math>\C,</math> the set of [[complex number]]s, and <math>\C^n</math> have a standard topology in which the basic open sets are open balls.

=== Topology from algebraic structure ===
For any [[Algebraic structure|algebraic objects]] we can introduce the discrete topology, under which the algebraic operations are continuous functions.  For any such structure that is not finite, we often have a natural topology compatible with the algebraic operations, in the sense that the algebraic operations are still continuous.  This leads to concepts such as [[topological group]]s, [[topological ring]]s, [[topological field]]s and [[topological vector space]]s over the latter. [[Local field]]s are topological fields important in [[number theory]].

The [[Zariski topology]] is defined algebraically on the [[spectrum of a ring]] or an [[algebraic variety]].  On <math>\R^n</math> or <math>\C^n,</math> the closed sets of the Zariski topology are the [[solution set]]s of systems of [[polynomial]] equations.

=== Topological spaces with order structure ===
* '''Spectral''': A space is ''[[Spectral space|spectral]]'' if and only if it is the prime [[spectrum of a ring]] ([[Melvin Hochster|Hochster]] theorem).
* '''Specialization preorder''':  In a space the [[Specialization preorder|''specialization preorder'' (or ''canonical preorder'')]] is defined by <math>x \leq y</math> if and only if <math>\operatorname{cl}\{ x \} \subseteq \operatorname{cl}\{ y \},</math> where <math>\operatorname{cl}</math> denotes an operator satisfying the [[Kuratowski closure axioms]].

=== Topology from other structure ===
If <math>\Gamma</math> is a [[Filter on a set|filter]] on a set <math>X</math> then <math>\{ \varnothing \} \cup \Gamma</math> is a topology on <math>X.</math>

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.

A [[linear graph]] has a natural topology that generalizes many of the geometric aspects of [[Graph theory|graph]]s with [[Vertex (graph theory)|vertices]] and [[Graph (discrete mathematics)#Graph|edges]].

[[Outer space (mathematics)|Outer space]] of a [[free group]] <math>F_n</math> consists of the so-called "marked metric graph structures" of volume 1 on <math>F_n.</math><ref name="CV86">{{cite journal|last1= Culler|first1= Marc|author-link= Marc Culler|last2= Vogtmann|first2= Karen|author-link2= Karen Vogtmann|title= Moduli of graphs and automorphisms of free groups|journal=[[Inventiones Mathematicae]]|volume= 84|issue= 1|pages= 91–119|date= 1986|url= https://www.math.cornell.edu/~vogtmann/ScannedPapers/1986.0084.pdf|doi= 10.1007/BF01388734 |bibcode= 1986InMat..84...91C|s2cid= 122869546}}</ref>

== Classification of topological spaces ==
{{main|Topological property}}
Topological spaces can be broadly classified, [[up to]] homeomorphism, by their [[topological properties]].  A topological property is a property of spaces that is invariant under homeomorphisms.  To prove that two spaces are not homeomorphic it is sufficient to find a topological property not shared by them.  Examples of such properties include [[Connectedness (topology)|connectedness]], [[Compactness (topology)|compactness]], and various [[separation axiom]]s. For algebraic invariants see [[algebraic topology]].

== See also ==

* [[Complete Heyting algebra]] – The system of all open sets of a given topological space ordered by inclusion is a complete Heyting algebra.
* {{annotated link|Compact space}}
* {{annotated link|Convergence space}}
* {{annotated link|Exterior space}}
* {{annotated link|Hausdorff space}}
* {{annotated link|Hilbert space}}
* {{annotated link|Hemicontinuity}}
* {{annotated link|Linear subspace}}
* [[Pointless topology]]
* {{annotated link|Quasitopological space}}
* {{annotated link|Relatively compact subspace}}
* {{annotated link|Space (mathematics)}}

== Citations ==
{{reflist}}

== Bibliography ==

* {{cite book|last=Armstrong|first=M. A.|title=Basic Topology|series=[[Undergraduate Texts in Mathematics]]|year=1983|orig-year=1979|publisher=Springer|isbn=0-387-90839-0 }}
* [[Glen Bredon|Bredon, Glen E.]], ''Topology and Geometry'' (Graduate Texts in Mathematics), Springer; 1st edition (October 17, 1997). {{isbn|0-387-97926-3}}.
* [[Nicolas Bourbaki|Bourbaki, Nicolas]]; ''Elements of Mathematics: General Topology'', Addison-Wesley (1966).
* {{cite book|author-link=Ronald Brown (mathematician)|last=Brown|first= Ronald|url=http://groupoids.org.uk/topgpds.html|title=Topology and Groupoids|publisher= Booksurge|year=2006|isbn=1-4196-2722-8}} (3rd edition of differently titled books) 
* [[Eduard Čech|Čech, Eduard]]; ''Point Sets'', Academic Press (1969).
* [[William Fulton (mathematician)|Fulton, William]], ''Algebraic Topology'', (Graduate Texts in Mathematics), Springer; 1st edition (September 5, 1997). {{isbn|0-387-94327-7}}.
* {{cite book|last1=Gallier|first1=Jean|last2=Xu|first2=Dianna|author2-link=Dianna Xu|title=A Guide to the Classification Theorem for Compact Surfaces|title-link=A Guide to the Classification Theorem for Compact Surfaces|date=2013|publisher=Springer}}
* {{cite book|last=Gauss|first= Carl Friedrich|title=General investigations of curved surfaces|year= 1827}}
* Lipschutz, Seymour; ''Schaum's Outline of General Topology'', McGraw-Hill; 1st edition (June 1, 1968). {{isbn|0-07-037988-2}}.
* [[James Munkres|Munkres, James]]; ''Topology'', Prentice Hall; 2nd edition (December 28, 1999). {{isbn|0-13-181629-2}}.
* Runde, Volker; ''A Taste of Topology (Universitext)'', Springer; 1st edition (July 6, 2005). {{isbn|0-387-25790-X}}.
* {{citation|first=Horst|last=Schubert|title=Topology|year=1968|publisher=Macdonald Technical & Scientific|isbn=0-356-02077-0}}
* [[Lynn Arthur Steen|Steen, Lynn A.]] and [[J. Arthur Seebach, Jr.|Seebach, J. Arthur Jr.]]; ''[[Counterexamples in Topology]]'', Holt, Rinehart and Winston (1970). {{isbn|0-03-079485-4}}.
* {{cite book|author=Vaidyanathaswamy, R.|author-link=Ramaswamy S. Vaidyanathaswamy|title=Set Topology|publisher=Chelsea Publishing Co.|year=1999|isbn=0486404560}}
* {{cite book|author=Willard, Stephen|title=General Topology|publisher=Dover Publications|year=2004|isbn=0-486-43479-6|url=https://books.google.com/books?id=-o8xJQ7Ag2cC&q=%22topological+space%22}}

== External links ==
{{Wikiquote}}
* {{Springer|title=Topological space|id=p/t093130}}

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[[Category:General topology]]
[[Category:Topological spaces| ]]