View source for Characterizations of the category of topological spaces

In [[Mathematics|mathematics]], a [[Topological space|topological space]] is usually defined in terms of [[Open set|open set]]s. However, there are many equivalent '''[[Characterization (mathematics)|characterization]]s of the [[Category of topological spaces|category of topological spaces]]'''. Each of these definitions provides a new way of thinking about topological concepts, and many of these have led to further lines of inquiry and generalisation.
== Definitions ==

Formally, each of the following definitions defines a [[Concrete category|concrete category]], and every pair of these categories can be shown to be ''concretely isomorphic''. This means that for every pair of categories defined below, there is an [[Isomorphism of categories|isomorphism of categories]], for which corresponding objects have the same [[Forgetful functor|underlying set]] and corresponding [[Morphism|morphism]]s are identical as set functions.

To actually establish the concrete isomorphisms is more tedious than illuminating. The simplest approach is probably to construct pairs of inverse concrete isomorphisms between each category and the [[Category of topological spaces|category of topological spaces]] '''Top'''. This would involve the following:

#Defining inverse object functions, checking that they are inverse, and checking that corresponding objects have the same underlying set.
#Checking that a set function is "continuous" (i.e., a morphism) in the given category [[If and only if|if and only if]] it is continuous (a morphism) in '''Top'''.

== Definition via open sets ==

''Objects'': all [[Topological space|topological space]]s, i.e., all pairs (''X'',''T'') of [[Set (mathematics)|set]] ''X'' together with a collection ''T'' of [[Subset|subset]]s of ''X'' satisfying:

# The [[Empty set|empty set]] and ''X'' are in ''T''.
# The [[Union (set theory)|union]] of any collection of sets in ''T'' is also in ''T''.
# The [[Intersection (set theory)|intersection]] of any pair of sets in ''T'' is also in ''T''.

:The sets in ''T'' are the '''[[Open set|open set]]s'''.

''Morphisms'': all ordinary continuous functions, i.e. all functions such that the inverse image of every open set is open.

''Comments'': This is the ordinary [[Category of topological spaces|category of topological spaces]].

== Definition via closed sets ==

''Objects'': all pairs (''X'',''T'') of [[Set (mathematics)|set]] ''X'' together with a collection ''T'' of [[Subset|subset]]s of ''X'' satisfying:

# The [[Empty set|empty set]] and ''X'' are in ''T''.
# The [[Intersection (set theory)|intersection]] of any collection of sets in ''T'' is also in ''T''.
# The [[Union (set theory)|union]] of any pair of sets in ''T'' is also in ''T''.

:The sets in ''T'' are the '''[[Closed set|closed set]]s'''.

''Morphisms'': all functions such that the inverse image of every closed set is closed.

''Comments'': This is the category that results by replacing each [[Lattice (order)|lattice]] of open sets in a topological space by its [[Duality (order theory)|order-theoretic dual]] of closed sets, the lattice of complements of open sets. The relation between the two definitions is given by [[De Morgan's laws]].

== Definition via closure operators ==

''Objects'': all pairs (''X'',cl) of set ''X'' together with a [[Closure operator|closure operator]] cl : ''P''(''X'') → ''P''(''X'') satisfying the [[Kuratowski closure axioms]]:

# <math> A \subseteq \operatorname{cl}(A) </math> (Extensivity)
# <math> \operatorname{cl}(\operatorname{cl}(A)) = \operatorname{cl}(A) </math> (Idempotence)
# <math> \operatorname{cl}(A \cup B) = \operatorname{cl}(A) \cup \operatorname{cl}(B) </math> (Preservation of binary unions)
# <math> \operatorname{cl}(\varnothing) = \varnothing </math> (Preservation of nullary unions)

''Morphisms'': all ''closure-preserving functions'', i.e., all functions ''f'' between two closure spaces

:<math>f\colon (X,\operatorname{cl}) \to (X',\operatorname{cl}')</math>

such that for all subsets <math>A</math> of <math>X</math>

:<math>f(\operatorname{cl}(A)) \subseteq \operatorname{cl}'(f(A))</math>

''Comments'': The Kuratowski closure axioms abstract the properties of the closure operator on a topological space, which assigns to each subset its topological closure. This topological [[Closure operator|closure operator]] has been generalized in [[Category theory|category theory]]; see ''Categorical Closure Operators'' by G. Castellini in "Categorical Perspectives", referenced below.

== Definition via a binary relation between points and subsets ==

Similar to the Kuratowski closure axioms approach, one can also define a topological space as a set <math>X</math> together with a relation <math>\prec</math> between points and subsets (<math>x\prec A</math> intuitively expresses that using the elements of <math>A</math> one can get arbitrarily close to <math>x</math>) satisfying
* There is no point <math>x\in X</math> such that <math>x\prec\emptyset</math>.
* If <math>a\in A</math>, then <math>a\prec A</math>.
* If <math>x\prec A\cup B</math>, then <math>x\prec A</math> or <math>x\prec B</math>.
* If each element <math>a\in A</math> satisfies <math>a\prec B</math> and <math>x\prec A</math>, then <math>x\prec B</math>.<ref>{{Cite web|url=https://mathoverflow.net/a/19173/103598|title=Big list - Why is a topology made up of 'open' sets?}}</ref>

== Definition via interior operators ==

''Objects'': all pairs (''X'',int) of set ''X'' together with an interior operator int : ''P''(''X'') → ''P''(''X'') satisfying the following [[Duality (order theory)|dualisation]] of the [[Kuratowski closure axioms]]:

# <math> A \supseteq \operatorname{int}(A) </math>
# <math> \operatorname{int}(\operatorname{int}(A)) = \operatorname{int}(A) </math> (Idempotence)
# <math> \operatorname{int}(A \cap B) = \operatorname{int}(A) \cap \operatorname{int}(B) </math> (Preservation of binary intersections)
# <math> \operatorname{int}(X) = X </math> (Preservation of nullary intersections)

''Morphisms'': all ''interior-preserving functions'', i.e., all functions ''f'' between two interior spaces

:<math>f\colon (X,\operatorname{int}) \to (X',\operatorname{int}')</math>

such that for all subsets <math>A</math> of <math>X'</math>

:<math>f^{-1}(\operatorname{int}'(A)) \subset \operatorname{int}(f^{-1}(A))</math>

''Comments'': The interior operator assigns to each subset its topological interior, in the same way the closure operator assigns to each subset its topological closure.

== Definition via neighbourhoods ==

''Objects'': all pairs (''X'',''N'') of set ''X'' together with a '''neighbourhood function''' ''N'' : ''X'' → ''F''(''X''), where ''F''(''X'') denotes the set of all [[Filter (mathematics)|filter]]s on ''X'', satisfying for every ''x'' in ''X'':

#If ''U'' is in ''N''(''x''), then ''x'' is in ''U''.
#If ''U'' is in ''N''(''x''), then there exists ''V'' in ''N''(''x'') such that ''U'' is in ''N''(''y'') for all ''y'' in ''V''.

''Morphisms'': all ''neighbourhood-preserving functions'', i.e., all functions ''f'' : (''X'', ''N'') → (''Y'', ''N''') such that if ''V'' is in ''N''(''f''(''x'')), then there exists ''U'' in ''N''(''x'') such that ''f''(''U'') is contained in ''V''. This is equivalent to asking that whenever ''V'' is in ''N''(''f''(''x'')), then ''f''<sup>&minus;1</sup>(''V'') is in ''N''(''x'').

''Comments'': This definition axiomatizes the notion of neighbourhood. We say that ''U'' is a neighbourhood of ''x'' if ''U'' is in ''N''(''x''). The open sets can be recovered by declaring a set to be open if it is a neighbourhood of each of its points; the final axiom then states that every neighbourhood contains an open set. These axioms (coupled with the [[Hausdorff space|Hausdorff condition]]) can be retraced to [[Biography:Felix Hausdorff|Felix Hausdorff]]'s original definition of a topological space in [[Grundzüge der Mengenlehre]].

== Definition via convergence ==

The category of topological spaces can also be defined via a convergence relation between [[Filter (mathematics)|filters]] on ''X'' and points of ''x''. This definition demonstrates that convergence of filters can be viewed as a fundamental topological notion. A topology in the usual sense can be recovered by declaring a set ''A'' to be closed if, whenever ''F'' is a filter on ''A'', then ''A'' contains all points to which ''F'' converges.

Similarly, the category of topological spaces can also be described via net convergence. As for filters, this definition shows that convergence of nets can be viewed as a fundamental topological notion. A topology in the usual sense can be recovered by declaring a set ''A'' to be closed if, whenever (''x''<sub>α</sub>) is a net on ''A'', then ''A'' contains all points to which (''x''<sub>α</sub>) converges.

== See also ==

* {{annotated link|Cauchy space}}
* {{annotated link|Convergence space}}
* {{annotated link|Filters in topology}}
* {{annotated link|Net (mathematics)}}
* {{annotated link|Sequential space}}
* {{annotated link|Topological space}}
* {{annotated link|Topology}}

== References ==
{{reflist}}
* Adámek, Jiří, Herrlich, Horst, & Strecker, George E. (1990). [http://katmat.math.uni-bremen.de/acc/acc.pdf ''Abstract and Concrete Categories'']. Originally publ. John Wiley & Sons. {{ISBN|0-471-60922-6}}. (now free on-line edition)
* Joshi, K. D., ''Introduction to General Topology'', New Age International, 1983, {{ISBN|0-85226-444-5}}
* Koslowsk and Melton, eds., ''Categorical Perspectives'', Birkhauser, 2001, {{ISBN|0-8176-4186-6}}
* Wyler, Oswald (1996). Convergence axioms for topology. ''Ann. N. Y. Acad. Sci.'' '''806''', 465-475

[[Category:General topology]]

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