View source for Axiomatic foundations of topological spaces

In the [[mathematics|mathematical]] field of [[topology]], a [[topological space]] is usually defined by declaring its [[open set]]s.{{sfnm|1a1=Dugundji|1y=1966|2a1=Engelking|2y=1977|3a1=Kelley|3y=1955}} However, this is not necessary, as there are many equivalent axiomatic foundations, each leading to exactly the same concept. For instance, a topological space determines a class of [[closed set]]s, of closure and interior operators, and of convergence of various types of objects. Each of these can instead be taken as the primary class of objects, with all of the others (including the class of open sets) directly determined from that new starting point. For example, in [[Kazimierz Kuratowski]]'s well-known textbook on [[point-set topology]], a topological space is defined as a set together with a certain type of "closure operator," and all other concepts are derived therefrom.{{sfnm|1a1=Kuratowski|1y=1966|1loc=p.38}} Likewise, the neighborhood-based axioms (in the context of [[Hausdorff space]]s) can be retraced to [[Felix Hausdorff]]'s original definition of a topological space in [[Grundzüge der Mengenlehre]].{{citation needed|date=December 2021}}

Many different textbooks use many different inter-dependences of concepts to develop point-set topology. The result is always the same collection of objects: open sets, closed sets, and so on. For many practical purposes, the question of which foundation is chosen is irrelevant, as long as the meaning and interrelation between objects (many of which are given in this article), which are the same regardless of choice of development, are understood. However, there are cases where it can be useful to have flexibility. For instance, there are various natural notions of [[convergence of measures]], and it is not immediately clear whether they arise from a topological structure or not. Such questions are greatly clarified by the topological axioms based on convergence.

==Standard definitions via open sets{{anchor|Open set characterization}}==

A topological space is a set <math>X</math> together with a collection <math>S</math> of [[subset]]s of <math>X</math> satisfying:{{sfnm|1a1=Dugundji|1y=1966|1loc=p.62|2a1=Engelking|2y=1977|2loc=p.11-12|3a1=Kelley|3y=1955|3loc=p.37|4a1=Kuratowski|4y=1966|4loc=p.45}}
* The [[empty set]] and <math>X</math> are in <math>S.</math>
* The [[Union (set theory)|union]] of any collection of sets in <math>S</math> is also in <math>S.</math>
* The [[Intersection (set theory)|intersection]] of any pair of sets in <math>S</math> is also in <math>S.</math> Equivalently, the intersection of any finite collection of sets in <math>S</math> is also in <math>S.</math>

Given a topological space <math>(X, S),</math> one refers to the elements of <math>S</math> as the '''open sets''' of <math>X,</math> and it is common only to refer to <math>S</math> in this way, or by the label '''topology'''. Then one makes the following secondary definitions:
* Given a second topological space <math>Y,</math> a function <math>f : X \to Y</math> is said to be '''continuous''' if and only if for every open subset <math>U</math> of <math>Y,</math> one has that <math>f^{-1}(U)</math> is an open subset of <math>X.</math>{{sfnm|1a1=Dugundji|1y=1966|1loc=p.79|2a1=Engelking|2y=1977|2loc=p.27-28|3a1=Kelley|3y=1955|3loc=p.85|4a1=Kuratowski|4y=1966|4loc=p.105}}
* A subset <math>C</math> of <math>X</math> is '''closed''' if and only if its complement <math>X \setminus C</math> is open.{{sfnm|1a1=Dugundji|1y=1966|1loc=p.68|2a1=Engelking|2y=1977|2loc=p.13|3a1=Kelley|3y=1955|3loc=p.40}}
* Given a subset <math>A</math> of <math>X,</math> the '''closure''' is the set of all points such that any open set containing such a point must intersect <math>A.</math>{{sfnm|1a1=Dugundji|1y=1966|1loc=p.69|2a1=Engelking|2y=1977|2loc=p.13}}
* Given a subset <math>A</math> of <math>X,</math> the '''interior''' is the union of all open sets contained in <math>A.</math>{{sfnm|1a1=Dugundji|1y=1966|1loc=p.71|2a1=Engelking|2y=1977|2loc=p.14|3a1=Kelley|3y=1955|3loc=p.44|4a1=Kuratowski|4y=1966|4loc=p.58}}
* Given an element <math>x</math> of <math>X,</math> one says that a subset <math>A</math> is a '''neighborhood''' of <math>x</math> if and only if <math>x</math> is contained in an open subset of <math>X</math> which is also a subset of <math>A.</math>{{sfnm|1a1=Kelley|1y=1955|1loc=p.38|2a1=Kuratowski|2y=1966|2loc=p.61}} Some textbooks use "neighborhood of <math>x</math>" to instead refer to an open set containing <math>x.</math>{{sfnm|1a1=Dugundji|1y=1966|1loc=p.63|2a1=Engelking|2y=1977|2loc=p.12}}
* One says that a '''net converges to a point''' <math>x</math> of <math>X</math> if for any open set <math>U</math> containing <math>x,</math> the net is eventually contained in <math>U.</math>{{sfnm|1a1=Dugundji|1y=1966|1loc=p.210|2a1=Engelking|2y=1977|2loc=p.49|3a1=Kelley|3y=1955|3loc=p.66|4a1=Kuratowski|4y=1966|4loc=p.203}}
* Given a set <math>X,</math> a [[Filter (set theory)|filter]] is a collection of nonempty subsets of <math>X</math> that is closed under finite intersection and under supersets.{{sfnm|1a1=Engelking|1y=1977|1loc=p.52|2a1=Kelley|2y=1955|2loc=p.83}} Some textbooks allow a filter to contain the empty set, and reserve the name "proper filter" for the case in which it is excluded.{{sfnm|1a1=Kuratowski|1y=1966|1loc=p.6}} A topology on <math>X</math> defines a notion of a '''filter converging to a point''' <math>x</math> of <math>X,</math> by requiring that any open set <math>U</math> containing <math>x</math> is an element of the filter.{{sfnm|1a1=Engelking|1y=1977|1loc=p.52|2a1=Kelley|2y=1955|2loc=p.83|3a1=Kuratowski|3y=1966|3loc=p.63}}
* Given a set <math>X,</math> a filterbase is a collection of nonempty subsets such that every two subsets intersect nontrivially and contain a third subset in the intersection.{{sfnm|1a1=Dugundji|1y=1966|1loc=211|2a1=Engelking|2y=1977|2loc=p.52}} Given a topology on <math>X,</math> one says that a filterbase converges to a point <math>x</math> if every neighborhood of <math>x</math> contains some element of the filterbase.{{sfnm|1a1=Dugundji|1y=1966|1loc=p.212|2a1=Engelking|2y=1977|2loc=p.52}}

==Definition via closed sets{{anchor|Closed set characterization}}==

Let <math>X</math> be a topological space. According to [[De Morgan's laws]], the collection <math>T</math> of closed sets satisfies the following properties:{{sfnm|1a1=Dugundji|1y=1966|1loc=p.69|2a1=Engelking|2y=1977|2loc=p.13|3a1=Kelley|3y=1955|3loc=p.40|4a1=Kuratowski|4y=1966|4loc=p.44}}
* The [[empty set]] and <math>X</math> are elements of <math>T</math>
* The [[Intersection (set theory)|intersection]] of any collection of sets in <math>T</math> is also in <math>T.</math>
* The [[Union (set theory)|union]] of any pair of sets in <math>T</math> is also in <math>T.</math>
Now suppose that <math>X</math> is only a set. Given any collection <math>T</math> of subsets of <math>X</math> which satisfy the above axioms, the corresponding set <math>\{U : X \setminus U \in T\}</math> is a topology on <math>X,</math> and it is the only topology on <math>X</math> for which <math>T</math> is the corresponding collection of closed sets.{{sfnm|1a1=Dugundji|1y=1966|1loc=p.74|2a1=Engelking|2y=1977|2loc=p.22|3a1=Kelley|3y=1955|3loc=p.40|4a1=Kuratowski|4y=1966|4loc=p.44}} This is to say that a topology can be defined by declaring the closed sets. As such, one can rephrase all definitions to be in terms of closed sets:
* Given a second topological space <math>Y,</math> a function <math>f : X \to Y</math> is continuous if and only if for every closed subset <math>U</math> of <math>Y,</math> the set <math>f^{-1}(U)</math> is closed as a subset of <math>X.</math>{{sfnm|1a1=Dugundji|1y=1966|1loc=p.79|2a1=Engelking|2y=1977|2loc=p.28|3a1=Kelley|3y=1955|3loc=p.86|4a1=Kuratowski|4y=1966|4loc=p.105}}
* a subset <math>C</math> of <math>X</math> is open if and only if its complement <math>X \setminus C</math> is closed.{{sfnm|1a1=Kelley|1y=1955|1loc=p.41}}
* given a subset <math>A</math> of <math>X,</math> the closure is the intersection of all closed sets containing <math>A.</math>{{sfnm|1a1=Dugundji|1y=1966|1loc=p.70|2a1=Engelking|2y=1977|2log=p.13|3a1=Kelley|3y=1955|3loc=p.42}}
* given a subset <math>A</math> of <math>X,</math> the interior is the complement of the intersection of all closed sets containing <math>X \setminus A.</math>

==Definition via closure operators{{anchor|Closure characterization}}==

Given a topological space <math>X,</math> the closure can be considered as a map <math>\wp(X) \to \wp(X),</math> where <math>\wp(X)</math> denotes the [[power set]] of <math>X.</math> One has the following [[Kuratowski closure axioms]]:{{sfnm|1a1=Dugundji|1y=1966|1loc=p.69-70|2a1=Engelking|2y=1977|2loc=p.14|3a1=Kelley|3y=1955|3loc=p.42-43}}
* <math>A \subseteq \operatorname{cl}(A)</math>
* <math>\operatorname{cl}(\operatorname{cl}(A)) = \operatorname{cl}(A)</math>
* <math>\operatorname{cl}(A \cup B) = \operatorname{cl}(A) \cup \operatorname{cl}(B)</math>
* <math>\operatorname{cl}(\varnothing) = \varnothing</math>
If <math>X</math> is a set equipped with a mapping satisfying the above properties, then the set of all possible outputs of cl satisfies the previous axioms for closed sets, and hence defines a topology; it is the unique topology whose associated closure operator coincides with the given cl.{{sfnm|1a1=Dugundji|1y=1966|1loc=p.73|2a1=Engelking|2y=1977|2loc=p.22|3a1=Kelley|3y=1955|3loc=p.43}} As before, it follows that on a topological space <math>X,</math> all definitions can be phrased in terms of the closure operator:
*Given a second topological space <math>Y,</math> a function <math>f : X \to Y</math> is continuous if and only if for every subset <math>A</math> of <math>X,</math> one has that the set <math>f(\operatorname{cl}(A))</math> is a subset of <math>\operatorname{cl}(f(A)).</math>{{sfnm|1a1=Dugundji|1y=1966|1loc=p.80|2a1=Engelking|2y=1977|2loc=p.28|3a1=Kelley|3y=1955|3loc=p.86|4a1=Kuratowski|4y=1966|4loc=p.105}}
* A subset <math>A</math> of <math>X</math> is open if and only if <math>\operatorname{cl}(X \setminus A) = X \setminus A.</math>{{sfnm|1a1=Kuratowski|1y=1966|1loc=p.43}}
* A subset <math>C</math> of <math>X</math> is closed if and only if <math>\operatorname{cl}(C) = C.</math>{{sfnm|1a1=Dugundji|1y=1966|1loc=p.69|2a1=Kelley|2y=1955|2loc=p.42|3a1=Kuratowski|3y=1966|3loc=p.43}}
* Given a subset <math>A</math> of <math>X,</math> the interior is the complement of <math>\operatorname{cl}(X \setminus A).</math>{{sfnm|1a1=Dugundji|1y=1966|1loc=p.71|2a1=Engelking|2y=1977|2loc=p.15|3a1=Kelley|3y=1955|3loc=p.44-45|4a1=Kuratowski|4y=1966|4loc=p.55}}

==Definition via interior operators{{anchor|Interior characterization}}==
{{See also|Interior (topology)}}
Given a topological space <math>X,</math> the interior can be considered as a map <math>\wp(X) \to \wp(X),</math> where <math>\wp(X)</math> denotes the [[power set]] of <math>X.</math> It satisfies the following conditions:{{sfnm|1a1=Engelking|1y=1977|1loc=p.15}}
* <math>\operatorname{int}(A) \subseteq A</math>
* <math>\operatorname{int}(\operatorname{int}(A)) = \operatorname{int}(A)</math>
* <math>\operatorname{int}(A \cap B) = \operatorname{int}(A) \cap \operatorname{int}(B)</math>
* <math>\operatorname{int}(X) = X</math>
If <math>X</math> is a set equipped with a mapping satisfying the above properties, then the set of all possible outputs of int satisfies the previous axioms for open sets, and hence defines a topology; it is the unique topology whose associated interior operator coincides with the given int.{{sfnm|1a1=Dugundji|1y=1966|1loc=p.74|2a1=Engelking|2y=1977|2loc=p.23}} It follows that on a topological space <math>X,</math> all definitions can be phrased in terms of the interior operator, for instance:
* Given topological spaces <math>X</math> and <math>Y,</math> a function <math>f : X \to Y</math> is continuous if and only if for every subset <math>B</math> of <math>Y,</math> one has that the set <math>f^{-1}(\operatorname{int}(B))</math> is a subset of <math>\operatorname{int}(f^{-1}(B)).</math>{{sfnm|1a1=Engelking|1y=1977|1loc=p.28|2a1=Kuratowski|2y=1966|2loc=p.103}}
* A set is open if and only if it equals its interior.{{sfnm|1a1=Dugundji|1y=1966|1loc=p.71|2a1=Kelley|2y=1955|2loc=p.44}}
* The closure of a set is the complement of the interior of its complement.{{sfnm|1a1=Kelley|1y=1955|1loc=p.44-45}}

==Definition via exterior operators{{anchor|Exterior characterization}}==
{{See also|Interior (topology)#Exterior of a set}}
Given a topological space <math>X,</math> the exterior can be considered as a map <math>\wp(X) \to \wp(X),</math> where <math>\wp(X)</math> denotes the [[power set]] of <math>X.</math> It satisfies the following conditions:<ref name=":0">{{Cite journal |last=Lei |first=Yinbin |last2=Zhang |first2=Jun |date=August 2019 |title=Generalizing Topological Set Operators |url=https://doi.org/10.1016/j.entcs.2019.07.016 |journal=Electronic Notes in Theoretical Computer Science |volume=345 |pages=63–76 |doi=10.1016/j.entcs.2019.07.016 |issn=1571-0661|doi-access=free }}</ref>
* <math>\operatorname{ext}(\varnothing) = X</math>
* <math> A \cap \operatorname{ext}(A) = \varnothing </math>
* <math>\operatorname{ext}(X \setminus \operatorname{ext}(A)) = \operatorname{ext}(A)</math>
* <math>\operatorname{ext}(A \cup B) = \operatorname{ext}(A) \cap \operatorname{ext}(B)</math>
If <math>X</math> is a set equipped with a mapping satisfying the above properties, then we can define the interior operator and vice versa. More precisely, if we define <math>\operatorname{int}_\operatorname{ext} : \wp(X) \to \wp(X), 
 \operatorname{int}_\operatorname{ext} = \operatorname{ext}(X \setminus A)</math>, <math>\operatorname{int}_\operatorname{ext}</math> satisfies the interior operator axioms, and hence defines a topology.<ref>{{Cite book |last=Bourbaki |first=Nicolas |title=Elements of mathematics. Chapters 1/4: 3. General topology Chapters 1 - 4 |date=1998 |publisher=Springer |isbn=978-3-540-64241-1 |edition=Softcover ed., [Nachdr.] - [1998] |location=Berlin Heidelberg}}</ref> Conversely, if we define <math>\operatorname{ext}_\operatorname{int} : \wp(X) \to \wp(X), 
 \operatorname{ext}_\operatorname{int} = \operatorname{int}(X \setminus A)</math>, <math>\operatorname{ext}_\operatorname{int}</math> satisfies the above axioms. Moreover, these correspondence is 1-1. It follows that on a topological space <math>X,</math> all definitions can be phrased in terms of the exterior operator, for instance:

* The closure of a set is the complement of its exterior, <math>\operatorname{cl}_\operatorname{ext} : \wp(X) \to \wp(X), 
 \operatorname{cl}_\operatorname{ext}(A) = X \setminus \operatorname{ext}(A)</math>.

*Given a second topological space <math>Y,</math> a function <math>f : X \to Y</math> is continuous if and only if for every subset <math>A</math> of <math>X,</math> one has that the set <math>f(X \setminus \operatorname{ext}(A))</math> is a subset of <math>X \setminus \operatorname{ext}(f(A)).</math> Equivalently, <math>f</math> is continuous if and only if for every subset <math>B</math> of <math>Y,</math> one has that the set <math>f^{-1}(\operatorname{ext}(X \setminus B))</math> is a subset of <math>\operatorname{ext}(X \setminus f^{-1}(B)).</math> 
* A set is open if and only if it equals the exterior of its complement.
* A set is closed if and only if it equals the complement of its exterior.

==Definition via boundary operators{{anchor|Boundary characterization}}==
{{See also|Boundary (topology)}}
Given a topological space <math>X,</math> the boundary can be considered as a map <math>\wp(X) \to \wp(X),</math> where <math>\wp(X)</math> denotes the [[power set]] of <math>X.</math> It satisfies the following conditions:<ref name=":0" />
* <math>\partial A = \partial(X \setminus A)</math>
* <math>\partial(\partial(A)) \subseteq \partial(A)</math>
* <math>\partial(A \cup B) \subseteq \partial(A) \cup \partial(B)</math>
* <math>A \subseteq B \Rightarrow \partial A \subseteq B \cup \partial B</math>
* <math>\partial(\varnothing) = \varnothing</math>
If <math>X</math> is a set equipped with a mapping satisfying the above properties, then we can define closure operator and vice versa. More precisely, if we define <math>\operatorname{cl}_\partial : \wp(X) \to \wp(X), 
 \operatorname{cl}_\partial(A) = A \cup \partial(A)</math>, <math>\operatorname{cl}_\partial</math> satisfies closure axioms, and hence boundary operation defines a topology. Conversely, if we define <math>\partial_\operatorname{cl} : \wp(X) \to \wp(X), 
\partial_\operatorname{cl} = \operatorname{cl}(A) \cap \operatorname{cl}(X \setminus A)</math>, <math>\partial_\operatorname{cl}</math> satisfies above axioms. Moreover, these correspondence is 1-1. It follows that on a topological space <math>X,</math> all definitions can be phrased in terms of the boundary operator, for instance:

* <math>\operatorname{int}_\partial : \wp(X) \to \wp(X), 
 \operatorname{int}_\partial(A) = A \setminus \partial(A)</math>
* A set is open if and only if <math>\partial(A) \cap A = \varnothing</math>.
* A set is closed if and only if <math>\partial(A) \subseteq A</math>.

== Definition via derived sets{{anchor|Boundary characterization}} ==
{{See also|Derived set (mathematics)}}

The '''derived set''' of a [[Set (mathematics)|subset]] <math>S</math> of a [[topological space]] <math>X</math> is the set of all points <math>x \in X</math> that are [[Limit point|limit points]] of <math>S,</math> that is, points <math>x</math> such that every [[Neighbourhood (mathematics)|neighbourhood]] of <math>x</math> contains a point of <math>S</math> other than <math>x</math> itself. The derived set of <math>S \subseteq X</math>, denoted <math>S^*</math>, satisfies the following conditions:<ref name=":0" />
* <math>\varnothing^* = \varnothing</math>
* For all <math>x \in X , x \not\in \{x\}^*</math>
* <math>A^{**} \subseteq A \cup A^*</math>
* <math>(A \cup B)^* = A^* \cup B^*</math>
Since a set <math>S</math> is closed if and only if <math>S^* \subseteq S </math>,<ref>{{Cite book |last=Baker |first=Crump W. |title=Introduction to topology |date=1991 |publisher=Wm. C. Brown Publishers |isbn=978-0-697-05972-7 |location=Dubuque, IA}}</ref> the derived set uniquely defines a topology. It follows that on a topological space <math>X,</math> all definitions can be phrased in terms of derived sets, for instance:

*<math>\operatorname{cl}(A) = A \cup A^*</math>.

*Given topological spaces <math>X</math> and <math>Y,</math> a function <math>f : X \to Y</math> is continuous if and only if for every subset <math>B</math> of <math>Y,</math> one has that the set <math>f(B^*)</math> is a subset of <math>{f(B)}^* \cup f(B)</math>.<ref>{{Cite book |last=Hocking |first=John G. |title=Topology |last2=Young |first2=Gail S. |date=1988 |publisher=Dover Publications |isbn=978-0-486-65676-2 |location=New York}}</ref>

==Definition via neighbourhoods{{anchor|Neighbourhood characterization|Neighborhood characterization|Neighbourhood characterization of topological spaces|Neighborhood characterization of topological spaces}}==
{{See also|Neighbourhood system|Neighbourhood (mathematics)}}

Recall that this article follows the convention that a [[Neighbourhood (mathematics)|neighborhood]] is not necessarily open. In a topological space, one has the following facts:{{sfn|Willard|2004|pp=31-32}}
* If <math>U</math> is a neighborhood of <math>x</math> then <math>x</math> is an element of <math>U.</math>
* The intersection of two neighborhoods of <math>x</math> is a neighborhood of <math>x.</math> Equivalently, the intersection of finitely many neighborhoods of <math>x</math> is a neighborhood of <math>x.</math>
* If <math>V</math> contains a neighborhood of <math>x,</math> then <math>V</math> is a neighborhood of <math>x.</math>
* If <math>U</math> is a neighborhood of <math>x,</math> then there exists a neighborhood <math>V</math> of <math>x</math> such that <math>U</math> is a neighborhood of each point of <math>V</math>.
If <math>X</math> is a set and one declares a nonempty collection of neighborhoods for every point of <math>X,</math> satisfying the above conditions, then a topology is defined by declaring a set to be open if and only if it is a neighborhood of each of its points; it is the unique topology whose associated system of neighborhoods is as given.{{sfn|Willard|2004|pp=31-32}} It follows that on a topological space <math>X,</math> all definitions can be phrased in terms of neighborhoods:
* Given another topological space <math>Y,</math> a map <math>f : X \to Y</math> is continuous if and only for every element <math>x</math> of <math>X</math> and every neighborhood <math>B</math> of <math>f(x),</math> the preimage <math>f^{-1}(B)</math> is a neighborhood of <math>x.</math>{{sfnm|1a1=Kuratowski|1y=1966|1loc=p.103}}
* A subset of <math>X</math> is open if and only if it is a neighborhood of each of its points.
* Given a subset <math>A</math> of <math>X,</math> the interior is the collection of all elements <math>x</math> of <math>X</math> such that <math>A</math> is a neighbourhood of <math>x</math>.
* Given a subset <math>A</math> of <math>X,</math> the closure is the collection of all elements <math>x</math> of <math>X</math> such that every neighborhood of <math>x</math> intersects <math>A.</math>{{sfnm|1a1=Kuratowski|1y=1966|1loc=p.61}}

==Definition via convergence of nets{{anchor|Net characterization|Convergent net characterization}}==
{{See also|Net (mathematics)|Fréchet–Urysohn space}}

Convergence of nets satisfies the following properties:{{sfnm|1a1=Kelley|1y=1955|1loc=p.74}}{{sfn|Willard|2004|p=77}}

# Every constant net converges to itself.
# Every [[Subnet (mathematics)|subnet]] of a convergent net converges to the same limits.
# If a net does not converge to a point <math>x</math> then there is a subnet such that no further subnet converges to <math>x.</math> Equivalently, if <math>x_{\bull}</math> is a net such that every one of its subnets has a sub-subnet that converges to a point <math>x,</math> then <math>x_{\bull}</math> converges to <math>x.</math>
# ''{{visible anchor|Diagonal principle}}''/''Convergence of iterated limits''. If <math>\left(x_a\right)_{a \in A} \to x</math> in <math>X</math> and for every index <math>a \in A,</math> <math>\left(x_a^i\right)_{i \in I_a}</math> is a net that [[Convergent net|converges to]] <math>x_a</math> in <math>X,</math> then there exists a diagonal (sub)net of <math>\left(x_a^i\right)_{a \in A, i \in I_a}</math> that converges to <math>x.</math>
::* A {{em|{{visible anchor|diagonal net}}}} refers to any [[Subnet (mathematics)|subnet]] of <math>\left(x_a^i\right)_{a \in A, i \in I_a}.</math>
::* The : notation <math>\left(x_a^i\right)_{a \in A, i \in I_a}</math> denotes the net defined by <math>(a, i) \mapsto x_a^i</math> whose domain is the set <math>{\textstyle\bigcup\limits_{a \in A}} A \times I_a</math> [[Lexicographic order|ordered lexicographically]] first by <math>A</math> and then by <math>I_a;</math>{{sfn|Willard|2004|p=77}} explicitly, given any two pairs <math>(a_1, i_1), \left(a_2, i_2\right) \in {\textstyle\bigcup\limits_{a \in A}} A \times I_a,</math> declare that <math>(a_1, i_1) \leq \left(a_2, i_2\right)</math> holds if and only if both (1) <math>a_1 \leq a_2,</math> and also (2) if <math>a_1 = a_2</math> then <math>i_1 \leq i_2.</math>


If <math>X</math> is a set, then given a notion of net convergence (telling what nets converge to what points{{sfn|Willard|2004|p=77}}) satisfying the above four axioms, a closure operator on <math>X</math> is defined by sending any given set <math>A</math> to the set of all limits of all nets valued in <math>A;</math> the corresponding topology is the unique topology inducing the given convergences of nets to points.{{sfnm|1a1=Kelley|1y=1955|1loc=p.74}} 

Given a subset <math>A \subseteq X</math> of a topological space <math>X:</math>
* <math>A</math> is open in <math>X</math> if and only if every net converging to an element of <math>A</math> is eventually contained in <math>A.</math>
* the closure of <math>A</math> in <math>X</math> is the set of all limits of all convergent nets valued in <math>A.</math>{{sfnm|1a1=Engelking|1y=1977|1loc=p.50|2a1=Kelley|2y=1955|2loc=p.66}}{{sfn|Willard|2004|p=77}}
* <math>A</math> is closed in <math>X</math> if and only if there does not exist a net in <math>A</math> that converges to an element of the complement <math>X \setminus A.</math>{{sfnm|1a1=Engelking|1y=1977|1loc=p.51|2a1=Kelley|2y=1955|2loc=p.66}} A subset <math>A \subseteq X</math> is closed in <math>X</math> if and only if every limit point of every convergent net in <math>A</math> necessarily belongs to <math>A.</math>{{sfn|Willard|2004|pp=73-77}} 

A function <math>f : X \to Y</math> between two topological spaces is continuous if and only if for every <math>x \in X</math> and every net <math>x_{\bull}</math> in <math>X</math> that [[Convergent net|converges to]] <math>x</math> in <math>X,</math> the net <math>f\left(x_{\bull}\right)</math><ref group=note>Assuming that the net <math>x_{\bull}</math> is indexed by <math>I</math> (so that <math>x_{\bull} = \left(x_i\right)_{i \in I},</math> which is just notation for function <math>x_\bull : I \to X</math> that sends <math>i \mapsto x_i</math>) then <math>f\left(x_{\bull}\right)</math> denotes the [[Function composition|composition]] of <math>x_\bull : I \to X</math> with <math>f : X \to Y.</math> That is, <math>f\left(x_{\bull}\right) := f \circ x_\bull = \left(f\left(x_i\right)\right)_{i \in I}</math> is the function <math>f \circ x_\bull : I \to Y.</math></ref> converges to <math>f(x)</math> in <math>Y.</math>{{sfnm|1a1=Engelking|1y=1977|1loc=p.51|2a1=Kelley|2y=1955|2loc=p.86}}

==Definition via convergence of filters{{anchor|Filter characterization|Convergent filter characterization}}==
{{See also|Filters in topology}}

A topology can also be defined on a set by declaring which filters converge to which points.{{citation needed|date=December 2021}} One has the following characterizations of standard objects in terms of [[Filter (set theory)|filters]] and [[prefilter]]s (also known as filterbases):
* Given a second topological space <math>Y,</math> a function <math>f : X \to Y</math> is continuous if and only if it preserves [[Convergent prefilter|convergence of prefilters]].{{sfnm|1a1=Dugundji|1y=1966|1loc=p.216|2a1=Engelking|2y=1977|2loc=p.52}}
* A subset <math>A</math> of <math>X</math> is open if and only if every [[Convergent filter|filter converging]] to an element of <math>A</math> contains <math>A.</math>{{sfnm|1a1=Kelley|1y=1955|1loc=p.83}}
* A subset <math>A</math> of <math>X</math> is closed if and only if there does not exist a prefilter on <math>A</math> which converges to a point in the complement <math>X \setminus A.</math>{{sfnm|1a1=Dugundji|1y=1966|1loc=p.215}}
* Given a subset <math>A</math> of <math>X,</math> the closure consists of all points <math>x</math> for which there is a prefilter on <math>A</math> converging to <math>x.</math>{{sfnm|1a1=Dugundji|1y=1966|1loc=p.215|2a1=Engelking|2y=1977|2loc=p.52}}
* A subset <math>A</math> of <math>X</math> is a neighborhood of <math>x</math> if and only if it is an element of every filter converging to <math>x.</math>{{sfnm|1a1=Kelley|1y=1955|1loc=p.83}}

==See also==

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

==Citations==

{{reflist|30em}}

'''Notes'''
{{reflist|group=note}}

==References==

* {{wikicite|ref={{sfnRef|Dugundji|1966}}|reference={{cite book|author-link=James Dugundji|last=Dugundji|first=James|title=Topology|edition=Reprinting of the 1966 original|series=Allyn and Bacon Series in Advanced Mathematics|publisher=Allyn and Bacon, Inc.|location=Boston, Mass.–London–Sydney|year=1978}}}}
* {{wikicite|reference={{cite book|author-link=Ryszard Engelking|last=Engelking|first=Ryszard|title=General topology|edition=Translated by author from Polish|series=Monografie Matematyczne|volume=60|publisher=PWN—Polish Scientific Publishers|location=Warsaw|year=1977}}}}
* {{wikicite|ref={{sfnRef|Kelley|1955}}|reference={{cite book|author-link=John L. Kelley|last=Kelley|first=John L.|title=General topology|edition=Reprint of the 1955|series=Graduate Texts in Mathematics|volume=27|publisher=Springer-Verlag|location=New York-Berlin|year=1975}}}}
* {{wikicite|reference={{cite book|author-link=Kazimierz Kuratowski|last=Kuratowski|first=K.|title=Topology. Vol. I.|edition=Translated from the French by J. Jaworowski. Revised and augmented|publisher=Academic Press/Państwowe Wydawnictwo Naukowe|location=New York-London/Warsaw|year=1966}}}}
* {{Willard General Topology}} <!--{{sfn|Willard|2004|p=}}-->

[[Category:Categories in category theory]]
[[Category:General topology]]