Axiomatic foundations of topological spaces

<h2 id="standard-definitions-via-open-sets">Standard definitions via open sets</h2>
A topological space is a set 
 
 
 
 X
 
 
 {\displaystyle X}
 
 together with a collection 
 
 
 
 S
 
 
 {\displaystyle S}
 
 of <a href="/facts/Subset/SJGERmbA">subsets</a> of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 satisfying:<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>

<ul><li>The <a href="/facts/Empty_set/ffEk3eA6">empty set</a> and 
 
 
 
 X
 
 
 {\displaystyle X}
 
 are in 
 
 
 
 S
 .
 
 
 {\displaystyle S.}
 
</li>
<li>The <a href="/facts/Union_(set_theory)/KVVXKfWl">union</a> of any collection of sets in 
 
 
 
 S
 
 
 {\displaystyle S}
 
 is also in 
 
 
 
 S
 .
 
 
 {\displaystyle S.}
 
</li>
<li>The <a href="/facts/Intersection_(set_theory)/RPfUNJh9">intersection</a> of any pair of sets in 
 
 
 
 S
 
 
 {\displaystyle S}
 
 is also in 
 
 
 
 S
 .
 
 
 {\displaystyle S.}
 
 Equivalently, the intersection of any finite collection of sets in 
 
 
 
 S
 
 
 {\displaystyle S}
 
 is also in 
 
 
 
 S
 .
 
 
 {\displaystyle S.}
 
</li></ul>
Given a topological space 
 
 
 
 (
 X
 ,
 S
 )
 ,
 
 
 {\displaystyle (X,S),}
 
 one refers to the elements of 
 
 
 
 S
 
 
 {\displaystyle S}
 
 as the open sets of 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 and it is common only to refer to 
 
 
 
 S
 
 
 {\displaystyle S}
 
 in this way, or by the label topology. Then one makes the following secondary definitions:

<ul><li>Given a second topological space 
 
 
 
 Y
 ,
 
 
 {\displaystyle Y,}
 
 a function 
 
 
 
 f
 :
 X
 →
 Y
 
 
 {\displaystyle f:X\to Y}
 
 is said to be continuous if and only if for every open subset 
 
 
 
 U
 
 
 {\displaystyle U}
 
 of 
 
 
 
 Y
 ,
 
 
 {\displaystyle Y,}
 
 one has that 
 
 
 
 
 f
 
 −
 1
 
 
 (
 U
 )
 
 
 {\displaystyle f^{-1}(U)}
 
 is an open subset of 
 
 
 
 X
 .
 
 
 {\displaystyle X.}
 
<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a></li>
<li>A subset 
 
 
 
 C
 
 
 {\displaystyle C}
 
 of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is closed if and only if its complement 
 
 
 
 X
 ∖
 C
 
 
 {\displaystyle X\setminus C}
 
 is open.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a></li>
<li>Given a subset 
 
 
 
 A
 
 
 {\displaystyle A}
 
 of 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 the closure is the set of all points such that any open set containing such a point must intersect 
 
 
 
 A
 .
 
 
 {\displaystyle A.}
 
<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a></li>
<li>Given a subset 
 
 
 
 A
 
 
 {\displaystyle A}
 
 of 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 the interior is the union of all open sets contained in 
 
 
 
 A
 .
 
 
 {\displaystyle A.}
 
<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a></li>
<li>Given an element 
 
 
 
 x
 
 
 {\displaystyle x}
 
 of 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 one says that a subset 
 
 
 
 A
 
 
 {\displaystyle A}
 
 is a neighborhood of 
 
 
 
 x
 
 
 {\displaystyle x}
 
 if and only if 
 
 
 
 x
 
 
 {\displaystyle x}
 
 is contained in an open subset of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 which is also a subset of 
 
 
 
 A
 .
 
 
 {\displaystyle A.}
 
<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a> Some textbooks use "neighborhood of 
 
 
 
 x
 
 
 {\displaystyle x}
 
" to instead refer to an open set containing 
 
 
 
 x
 .
 
 
 {\displaystyle x.}
 
<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a></li>
<li>One says that a net converges to a point 
 
 
 
 x
 
 
 {\displaystyle x}
 
 of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 if for any open set 
 
 
 
 U
 
 
 {\displaystyle U}
 
 containing 
 
 
 
 x
 ,
 
 
 {\displaystyle x,}
 
 the net is eventually contained in 
 
 
 
 U
 .
 
 
 {\displaystyle U.}
 
<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a></li>
<li>Given a set 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 a <a href="/facts/Filter_(set_theory)/7f3zk55u">filter</a> is a collection of nonempty subsets of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 that is closed under finite intersection and under supersets.<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a> Some textbooks allow a filter to contain the empty set, and reserve the name "proper filter" for the case in which it is excluded.<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a> A topology on 
 
 
 
 X
 
 
 {\displaystyle X}
 
 defines a notion of a filter converging to a point 
 
 
 
 x
 
 
 {\displaystyle x}
 
 of 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 by requiring that any open set 
 
 
 
 U
 
 
 {\displaystyle U}
 
 containing 
 
 
 
 x
 
 
 {\displaystyle x}
 
 is an element of the filter.<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a></li>
<li>Given a set 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 a filterbase is a collection of nonempty subsets such that every two subsets intersect nontrivially and contain a third subset in the intersection.<a class="footnote-ref" id="fnref:14" href="#fn:14">14</a> Given a topology on 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 one says that a filterbase converges to a point 
 
 
 
 x
 
 
 {\displaystyle x}
 
 if every neighborhood of 
 
 
 
 x
 
 
 {\displaystyle x}
 
 contains some element of the filterbase.<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a></li></ul>
<h2 id="definition-via-closed-sets">Definition via closed sets</h2>
Let 
 
 
 
 X
 
 
 {\displaystyle X}
 
 be a topological space. According to <a href="/facts/De_Morgan%2527s_laws/dj3F8xpt">De Morgan's laws</a>, the collection 
 
 
 
 T
 
 
 {\displaystyle T}
 
 of closed sets satisfies the following properties:<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a>

<ul><li>The <a href="/facts/Empty_set/ffEk3eA6">empty set</a> and 
 
 
 
 X
 
 
 {\displaystyle X}
 
 are elements of 
 
 
 
 T
 
 
 {\displaystyle T}
 
</li>
<li>The <a href="/facts/Intersection_(set_theory)/RPfUNJh9">intersection</a> of any collection of sets in 
 
 
 
 T
 
 
 {\displaystyle T}
 
 is also in 
 
 
 
 T
 .
 
 
 {\displaystyle T.}
 
</li>
<li>The <a href="/facts/Union_(set_theory)/KVVXKfWl">union</a> of any pair of sets in 
 
 
 
 T
 
 
 {\displaystyle T}
 
 is also in 
 
 
 
 T
 .
 
 
 {\displaystyle T.}
 
</li></ul>
Now suppose that 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is only a set. Given any collection 
 
 
 
 T
 
 
 {\displaystyle T}
 
 of subsets of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 which satisfy the above axioms, the corresponding set 
 
 
 
 {
 U
 :
 X
 ∖
 U
 ∈
 T
 }
 
 
 {\displaystyle \{U:X\setminus U\in T\}}
 
 is a topology on 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 and it is the only topology on 
 
 
 
 X
 
 
 {\displaystyle X}
 
 for which 
 
 
 
 T
 
 
 {\displaystyle T}
 
 is the corresponding collection of closed sets.<a class="footnote-ref" id="fnref:17" href="#fn:17">17</a> 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:

<ul><li>Given a second topological space 
 
 
 
 Y
 ,
 
 
 {\displaystyle Y,}
 
 a function 
 
 
 
 f
 :
 X
 →
 Y
 
 
 {\displaystyle f:X\to Y}
 
 is continuous if and only if for every closed subset 
 
 
 
 U
 
 
 {\displaystyle U}
 
 of 
 
 
 
 Y
 ,
 
 
 {\displaystyle Y,}
 
 the set 
 
 
 
 
 f
 
 −
 1
 
 
 (
 U
 )
 
 
 {\displaystyle f^{-1}(U)}
 
 is closed as a subset of 
 
 
 
 X
 .
 
 
 {\displaystyle X.}
 
<a class="footnote-ref" id="fnref:18" href="#fn:18">18</a></li>
<li>a subset 
 
 
 
 C
 
 
 {\displaystyle C}
 
 of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is open if and only if its complement 
 
 
 
 X
 ∖
 C
 
 
 {\displaystyle X\setminus C}
 
 is closed.<a class="footnote-ref" id="fnref:19" href="#fn:19">19</a></li>
<li>given a subset 
 
 
 
 A
 
 
 {\displaystyle A}
 
 of 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 the closure is the intersection of all closed sets containing 
 
 
 
 A
 .
 
 
 {\displaystyle A.}
 
<a class="footnote-ref" id="fnref:20" href="#fn:20">20</a></li>
<li>given a subset 
 
 
 
 A
 
 
 {\displaystyle A}
 
 of 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 the interior is the complement of the intersection of all closed sets containing 
 
 
 
 X
 ∖
 A
 .
 
 
 {\displaystyle X\setminus A.}
 
</li></ul>
<h2 id="definition-via-closure-operators">Definition via closure operators</h2>
Given a topological space 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 the closure can be considered as a map 
 
 
 
 ℘
 (
 X
 )
 →
 ℘
 (
 X
 )
 ,
 
 
 {\displaystyle \wp (X)\to \wp (X),}
 
 where 
 
 
 
 ℘
 (
 X
 )
 
 
 {\displaystyle \wp (X)}
 
 denotes the <a href="/facts/Power_set/FVuf8PDD">power set</a> of 
 
 
 
 X
 .
 
 
 {\displaystyle X.}
 
 One has the following <a href="/facts/Kuratowski_closure_axioms/ovjzpSms">Kuratowski closure axioms</a>:<a class="footnote-ref" id="fnref:21" href="#fn:21">21</a>

<ul><li>
 
 
 
 A
 ⊆
 cl
 ⁡
 (
 A
 )
 
 
 {\displaystyle A\subseteq \operatorname {cl} (A)}
 
</li>
<li>
 
 
 
 cl
 ⁡
 (
 cl
 ⁡
 (
 A
 )
 )
 =
 cl
 ⁡
 (
 A
 )
 
 
 {\displaystyle \operatorname {cl} (\operatorname {cl} (A))=\operatorname {cl} (A)}
 
</li>
<li>
 
 
 
 cl
 ⁡
 (
 A
 ∪
 B
 )
 =
 cl
 ⁡
 (
 A
 )
 ∪
 cl
 ⁡
 (
 B
 )
 
 
 {\displaystyle \operatorname {cl} (A\cup B)=\operatorname {cl} (A)\cup \operatorname {cl} (B)}
 
</li>
<li>
 
 
 
 cl
 ⁡
 (
 ∅
 )
 =
 ∅
 
 
 {\displaystyle \operatorname {cl} (\varnothing )=\varnothing }
 
</li></ul>
If 
 
 
 
 X
 
 
 {\displaystyle X}
 
 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.<a class="footnote-ref" id="fnref:22" href="#fn:22">22</a> As before, it follows that on a topological space 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 all definitions can be phrased in terms of the closure operator:

<ul><li>Given a second topological space 
 
 
 
 Y
 ,
 
 
 {\displaystyle Y,}
 
 a function 
 
 
 
 f
 :
 X
 →
 Y
 
 
 {\displaystyle f:X\to Y}
 
 is continuous if and only if for every subset 
 
 
 
 A
 
 
 {\displaystyle A}
 
 of 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 one has that the set 
 
 
 
 f
 (
 cl
 ⁡
 (
 A
 )
 )
 
 
 {\displaystyle f(\operatorname {cl} (A))}
 
 is a subset of 
 
 
 
 cl
 ⁡
 (
 f
 (
 A
 )
 )
 .
 
 
 {\displaystyle \operatorname {cl} (f(A)).}
 
<a class="footnote-ref" id="fnref:23" href="#fn:23">23</a></li>
<li>A subset 
 
 
 
 A
 
 
 {\displaystyle A}
 
 of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is open if and only if 
 
 
 
 cl
 ⁡
 (
 X
 ∖
 A
 )
 =
 X
 ∖
 A
 .
 
 
 {\displaystyle \operatorname {cl} (X\setminus A)=X\setminus A.}
 
<a class="footnote-ref" id="fnref:24" href="#fn:24">24</a></li>
<li>A subset 
 
 
 
 C
 
 
 {\displaystyle C}
 
 of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is closed if and only if 
 
 
 
 cl
 ⁡
 (
 C
 )
 =
 C
 .
 
 
 {\displaystyle \operatorname {cl} (C)=C.}
 
<a class="footnote-ref" id="fnref:25" href="#fn:25">25</a></li>
<li>Given a subset 
 
 
 
 A
 
 
 {\displaystyle A}
 
 of 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 the interior is the complement of 
 
 
 
 cl
 ⁡
 (
 X
 ∖
 A
 )
 .
 
 
 {\displaystyle \operatorname {cl} (X\setminus A).}
 
<a class="footnote-ref" id="fnref:26" href="#fn:26">26</a></li></ul>
<h2 id="definition-via-interior-operators">Definition via interior operators</h2>
See also: <a href="/facts/Interior_(topology)/5sxGnQfY">Interior (topology)</a>
Given a topological space 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 the interior can be considered as a map 
 
 
 
 ℘
 (
 X
 )
 →
 ℘
 (
 X
 )
 ,
 
 
 {\displaystyle \wp (X)\to \wp (X),}
 
 where 
 
 
 
 ℘
 (
 X
 )
 
 
 {\displaystyle \wp (X)}
 
 denotes the <a href="/facts/Power_set/FVuf8PDD">power set</a> of 
 
 
 
 X
 .
 
 
 {\displaystyle X.}
 
 It satisfies the following conditions:<a class="footnote-ref" id="fnref:27" href="#fn:27">27</a>

<ul><li>
 
 
 
 int
 ⁡
 (
 A
 )
 ⊆
 A
 
 
 {\displaystyle \operatorname {int} (A)\subseteq A}
 
</li>
<li>
 
 
 
 int
 ⁡
 (
 int
 ⁡
 (
 A
 )
 )
 =
 int
 ⁡
 (
 A
 )
 
 
 {\displaystyle \operatorname {int} (\operatorname {int} (A))=\operatorname {int} (A)}
 
</li>
<li>
 
 
 
 int
 ⁡
 (
 A
 ∩
 B
 )
 =
 int
 ⁡
 (
 A
 )
 ∩
 int
 ⁡
 (
 B
 )
 
 
 {\displaystyle \operatorname {int} (A\cap B)=\operatorname {int} (A)\cap \operatorname {int} (B)}
 
</li>
<li>
 
 
 
 int
 ⁡
 (
 X
 )
 =
 X
 
 
 {\displaystyle \operatorname {int} (X)=X}
 
</li></ul>
If 
 
 
 
 X
 
 
 {\displaystyle X}
 
 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.<a class="footnote-ref" id="fnref:28" href="#fn:28">28</a> It follows that on a topological space 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 all definitions can be phrased in terms of the interior operator, for instance:

<ul><li>Given topological spaces 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 ,
 
 
 {\displaystyle Y,}
 
 a function 
 
 
 
 f
 :
 X
 →
 Y
 
 
 {\displaystyle f:X\to Y}
 
 is continuous if and only if for every subset 
 
 
 
 B
 
 
 {\displaystyle B}
 
 of 
 
 
 
 Y
 ,
 
 
 {\displaystyle Y,}
 
 one has that the set 
 
 
 
 
 f
 
 −
 1
 
 
 (
 int
 ⁡
 (
 B
 )
 )
 
 
 {\displaystyle f^{-1}(\operatorname {int} (B))}
 
 is a subset of 
 
 
 
 int
 ⁡
 (
 
 f
 
 −
 1
 
 
 (
 B
 )
 )
 .
 
 
 {\displaystyle \operatorname {int} (f^{-1}(B)).}
 
<a class="footnote-ref" id="fnref:29" href="#fn:29">29</a></li>
<li>A set is open if and only if it equals its interior.<a class="footnote-ref" id="fnref:30" href="#fn:30">30</a></li>
<li>The closure of a set is the complement of the interior of its complement.<a class="footnote-ref" id="fnref:31" href="#fn:31">31</a></li></ul>
<h2 id="definition-via-exterior-operators">Definition via exterior operators</h2>
See also: <a href="/facts/Interior_(topology)/5sxGnQfY">Interior (topology) § Exterior of a set</a>
Given a topological space 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 the exterior can be considered as a map 
 
 
 
 ℘
 (
 X
 )
 →
 ℘
 (
 X
 )
 ,
 
 
 {\displaystyle \wp (X)\to \wp (X),}
 
 where 
 
 
 
 ℘
 (
 X
 )
 
 
 {\displaystyle \wp (X)}
 
 denotes the <a href="/facts/Power_set/FVuf8PDD">power set</a> of 
 
 
 
 X
 .
 
 
 {\displaystyle X.}
 
 It satisfies the following conditions:<a class="footnote-ref" id="fnref:32" href="#fn:32">32</a>

<ul><li>
 
 
 
 ext
 ⁡
 (
 ∅
 )
 =
 X
 
 
 {\displaystyle \operatorname {ext} (\varnothing )=X}
 
</li>
<li>
 
 
 
 A
 ∩
 ext
 ⁡
 (
 A
 )
 =
 ∅
 
 
 {\displaystyle A\cap \operatorname {ext} (A)=\varnothing }
 
</li>
<li>
 
 
 
 ext
 ⁡
 (
 X
 ∖
 ext
 ⁡
 (
 A
 )
 )
 =
 ext
 ⁡
 (
 A
 )
 
 
 {\displaystyle \operatorname {ext} (X\setminus \operatorname {ext} (A))=\operatorname {ext} (A)}
 
</li>
<li>
 
 
 
 ext
 ⁡
 (
 A
 ∪
 B
 )
 =
 ext
 ⁡
 (
 A
 )
 ∩
 ext
 ⁡
 (
 B
 )
 
 
 {\displaystyle \operatorname {ext} (A\cup B)=\operatorname {ext} (A)\cap \operatorname {ext} (B)}
 
</li></ul>
If 
 
 
 
 X
 
 
 {\displaystyle X}
 
 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 
 
 
 
 
 int
 
 ext
 
 
 :
 ℘
 (
 X
 )
 →
 ℘
 (
 X
 )
 ,
 
 int
 
 ext
 
 
 =
 ext
 ⁡
 (
 X
 ∖
 A
 )
 
 
 {\displaystyle \operatorname {int} _{\operatorname {ext} }:\wp (X)\to \wp (X),\operatorname {int} _{\operatorname {ext} }=\operatorname {ext} (X\setminus A)}
 
, 
 
 
 
 
 int
 
 ext
 
 
 
 
 {\displaystyle \operatorname {int} _{\operatorname {ext} }}
 
 satisfies the interior operator axioms, and hence defines a topology.<a class="footnote-ref" id="fnref:33" href="#fn:33">33</a> Conversely, if we define 
 
 
 
 
 ext
 
 int
 
 
 :
 ℘
 (
 X
 )
 →
 ℘
 (
 X
 )
 ,
 
 ext
 
 int
 
 
 =
 int
 ⁡
 (
 X
 ∖
 A
 )
 
 
 {\displaystyle \operatorname {ext} _{\operatorname {int} }:\wp (X)\to \wp (X),\operatorname {ext} _{\operatorname {int} }=\operatorname {int} (X\setminus A)}
 
, 
 
 
 
 
 ext
 
 int
 
 
 
 
 {\displaystyle \operatorname {ext} _{\operatorname {int} }}
 
 satisfies the above axioms. Moreover, these correspondence is 1-1. It follows that on a topological space 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 all definitions can be phrased in terms of the exterior operator, for instance:

<ul><li>The closure of a set is the complement of its exterior, 
 
 
 
 
 cl
 
 ext
 
 
 :
 ℘
 (
 X
 )
 →
 ℘
 (
 X
 )
 ,
 
 cl
 
 ext
 
 
 ⁡
 (
 A
 )
 =
 X
 ∖
 ext
 ⁡
 (
 A
 )
 
 
 {\displaystyle \operatorname {cl} _{\operatorname {ext} }:\wp (X)\to \wp (X),\operatorname {cl} _{\operatorname {ext} }(A)=X\setminus \operatorname {ext} (A)}
 
.</li></ul>
<ul><li>Given a second topological space 
 
 
 
 Y
 ,
 
 
 {\displaystyle Y,}
 
 a function 
 
 
 
 f
 :
 X
 →
 Y
 
 
 {\displaystyle f:X\to Y}
 
 is continuous if and only if for every subset 
 
 
 
 A
 
 
 {\displaystyle A}
 
 of 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 one has that the set 
 
 
 
 f
 (
 X
 ∖
 ext
 ⁡
 (
 A
 )
 )
 
 
 {\displaystyle f(X\setminus \operatorname {ext} (A))}
 
 is a subset of 
 
 
 
 X
 ∖
 ext
 ⁡
 (
 f
 (
 A
 )
 )
 .
 
 
 {\displaystyle X\setminus \operatorname {ext} (f(A)).}
 
 Equivalently, 
 
 
 
 f
 
 
 {\displaystyle f}
 
 is continuous if and only if for every subset 
 
 
 
 B
 
 
 {\displaystyle B}
 
 of 
 
 
 
 Y
 ,
 
 
 {\displaystyle Y,}
 
 one has that the set 
 
 
 
 
 f
 
 −
 1
 
 
 (
 ext
 ⁡
 (
 X
 ∖
 B
 )
 )
 
 
 {\displaystyle f^{-1}(\operatorname {ext} (X\setminus B))}
 
 is a subset of 
 
 
 
 ext
 ⁡
 (
 X
 ∖
 
 f
 
 −
 1
 
 
 (
 B
 )
 )
 .
 
 
 {\displaystyle \operatorname {ext} (X\setminus f^{-1}(B)).}
 
</li>
<li>A set is open if and only if it equals the exterior of its complement.</li>
<li>A set is closed if and only if it equals the complement of its exterior.</li></ul>
<h2 id="definition-via-boundary-operators">Definition via boundary operators</h2>
See also: <a href="/facts/Boundary_(topology)/WgdH18kp">Boundary (topology)</a>
Given a topological space 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 the boundary can be considered as a map 
 
 
 
 ℘
 (
 X
 )
 →
 ℘
 (
 X
 )
 ,
 
 
 {\displaystyle \wp (X)\to \wp (X),}
 
 where 
 
 
 
 ℘
 (
 X
 )
 
 
 {\displaystyle \wp (X)}
 
 denotes the <a href="/facts/Power_set/FVuf8PDD">power set</a> of 
 
 
 
 X
 .
 
 
 {\displaystyle X.}
 
 It satisfies the following conditions:<a class="footnote-ref" id="fnref:34" href="#fn:34">34</a>

<ul><li>
 
 
 
 ∂
 A
 =
 ∂
 (
 X
 ∖
 A
 )
 
 
 {\displaystyle \partial A=\partial (X\setminus A)}
 
</li>
<li>
 
 
 
 ∂
 (
 ∂
 (
 A
 )
 )
 ⊆
 ∂
 (
 A
 )
 
 
 {\displaystyle \partial (\partial (A))\subseteq \partial (A)}
 
</li>
<li>
 
 
 
 ∂
 (
 A
 ∪
 B
 )
 ⊆
 ∂
 (
 A
 )
 ∪
 ∂
 (
 B
 )
 
 
 {\displaystyle \partial (A\cup B)\subseteq \partial (A)\cup \partial (B)}
 
</li>
<li>
 
 
 
 A
 ⊆
 B
 ⇒
 ∂
 A
 ⊆
 B
 ∪
 ∂
 B
 
 
 {\displaystyle A\subseteq B\Rightarrow \partial A\subseteq B\cup \partial B}
 
</li>
<li>
 
 
 
 ∂
 (
 ∅
 )
 =
 ∅
 
 
 {\displaystyle \partial (\varnothing )=\varnothing }
 
</li></ul>
If 
 
 
 
 X
 
 
 {\displaystyle X}
 
 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 
 
 
 
 
 cl
 
 ∂
 
 
 :
 ℘
 (
 X
 )
 →
 ℘
 (
 X
 )
 ,
 
 cl
 
 ∂
 
 
 ⁡
 (
 A
 )
 =
 A
 ∪
 ∂
 (
 A
 )
 
 
 {\displaystyle \operatorname {cl} _{\partial }:\wp (X)\to \wp (X),\operatorname {cl} _{\partial }(A)=A\cup \partial (A)}
 
, 
 
 
 
 
 cl
 
 ∂
 
 
 
 
 {\displaystyle \operatorname {cl} _{\partial }}
 
 satisfies closure axioms, and hence boundary operation defines a topology. Conversely, if we define 
 
 
 
 
 ∂
 
 cl
 
 
 :
 ℘
 (
 X
 )
 →
 ℘
 (
 X
 )
 ,
 
 ∂
 
 cl
 
 
 =
 cl
 ⁡
 (
 A
 )
 ∩
 cl
 ⁡
 (
 X
 ∖
 A
 )
 
 
 {\displaystyle \partial _{\operatorname {cl} }:\wp (X)\to \wp (X),\partial _{\operatorname {cl} }=\operatorname {cl} (A)\cap \operatorname {cl} (X\setminus A)}
 
, 
 
 
 
 
 ∂
 
 cl
 
 
 
 
 {\displaystyle \partial _{\operatorname {cl} }}
 
 satisfies above axioms. Moreover, these correspondence is 1-1. It follows that on a topological space 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 all definitions can be phrased in terms of the boundary operator, for instance:

<ul><li>
 
 
 
 
 int
 
 ∂
 
 
 :
 ℘
 (
 X
 )
 →
 ℘
 (
 X
 )
 ,
 
 int
 
 ∂
 
 
 ⁡
 (
 A
 )
 =
 A
 ∖
 ∂
 (
 A
 )
 
 
 {\displaystyle \operatorname {int} _{\partial }:\wp (X)\to \wp (X),\operatorname {int} _{\partial }(A)=A\setminus \partial (A)}
 
</li>
<li>A set is open if and only if 
 
 
 
 ∂
 (
 A
 )
 ∩
 A
 =
 ∅
 
 
 {\displaystyle \partial (A)\cap A=\varnothing }
 
.</li>
<li>A set is closed if and only if 
 
 
 
 ∂
 (
 A
 )
 ⊆
 A
 
 
 {\displaystyle \partial (A)\subseteq A}
 
.</li></ul>
<h2 id="definition-via-derived-sets">Definition via derived sets</h2>
See also: <a href="/facts/Derived_set_(mathematics)/SqeHgT2t">Derived set (mathematics)</a>
The derived set of a <a href="/facts/Set_(mathematics)/BfucfHMq">subset</a> 
 
 
 
 S
 
 
 {\displaystyle S}
 
 of a <a href="/facts/Topological_space/v2ut7CbK">topological space</a> 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is the set of all points 
 
 
 
 x
 ∈
 X
 
 
 {\displaystyle x\in X}
 
 that are <a href="/facts/Limit_point/95MykoGU">limit points</a> of 
 
 
 
 S
 ,
 
 
 {\displaystyle S,}
 
 that is, points 
 
 
 
 x
 
 
 {\displaystyle x}
 
 such that every <a href="/facts/Neighbourhood_(mathematics)/XHKgrpkp">neighbourhood</a> of 
 
 
 
 x
 
 
 {\displaystyle x}
 
 contains a point of 
 
 
 
 S
 
 
 {\displaystyle S}
 
 other than 
 
 
 
 x
 
 
 {\displaystyle x}
 
 itself. The derived set of 
 
 
 
 S
 ⊆
 X
 
 
 {\displaystyle S\subseteq X}
 
, denoted 
 
 
 
 
 S
 
 ∗
 
 
 
 
 {\displaystyle S^{*}}
 
, satisfies the following conditions:<a class="footnote-ref" id="fnref:35" href="#fn:35">35</a>

<ul><li>
 
 
 
 
 ∅
 
 ∗
 
 
 =
 ∅
 
 
 {\displaystyle \varnothing ^{*}=\varnothing }
 
</li>
<li>For all 
 
 
 
 x
 ∈
 X
 ,
 x
 ∉
 {
 x
 
 }
 
 ∗
 
 
 
 
 {\displaystyle x\in X,x\not \in \{x\}^{*}}
 
</li>
<li>
 
 
 
 
 A
 
 ∗
 ∗
 
 
 ⊆
 A
 ∪
 
 A
 
 ∗
 
 
 
 
 {\displaystyle A^{**}\subseteq A\cup A^{*}}
 
</li>
<li>
 
 
 
 (
 A
 ∪
 B
 
 )
 
 ∗
 
 
 =
 
 A
 
 ∗
 
 
 ∪
 
 B
 
 ∗
 
 
 
 
 {\displaystyle (A\cup B)^{*}=A^{*}\cup B^{*}}
 
</li></ul>
Since a set 
 
 
 
 S
 
 
 {\displaystyle S}
 
 is closed if and only if 
 
 
 
 
 S
 
 ∗
 
 
 ⊆
 S
 
 
 {\displaystyle S^{*}\subseteq S}
 
,<a class="footnote-ref" id="fnref:36" href="#fn:36">36</a> the derived set uniquely defines a topology. It follows that on a topological space 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 all definitions can be phrased in terms of derived sets, for instance:

<ul><li>
 
 
 
 cl
 ⁡
 (
 A
 )
 =
 A
 ∪
 
 A
 
 ∗
 
 
 
 
 {\displaystyle \operatorname {cl} (A)=A\cup A^{*}}
 
.</li></ul>
<ul><li>Given topological spaces 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 ,
 
 
 {\displaystyle Y,}
 
 a function 
 
 
 
 f
 :
 X
 →
 Y
 
 
 {\displaystyle f:X\to Y}
 
 is continuous if and only if for every subset 
 
 
 
 B
 
 
 {\displaystyle B}
 
 of 
 
 
 
 Y
 ,
 
 
 {\displaystyle Y,}
 
 one has that the set 
 
 
 
 f
 (
 
 B
 
 ∗
 
 
 )
 
 
 {\displaystyle f(B^{*})}
 
 is a subset of 
 
 
 
 
 
 f
 (
 B
 )
 
 
 ∗
 
 
 ∪
 f
 (
 B
 )
 
 
 {\displaystyle {f(B)}^{*}\cup f(B)}
 
.<a class="footnote-ref" id="fnref:37" href="#fn:37">37</a></li></ul>
<h2 id="definition-via-neighbourhoods">Definition via neighbourhoods</h2>
See also: <a href="/facts/Neighbourhood_system/fMRM8Xht">Neighbourhood system</a> and <a href="/facts/Neighbourhood_(mathematics)/XHKgrpkp">Neighbourhood (mathematics)</a>
Recall that this article follows the convention that a <a href="/facts/Neighbourhood_(mathematics)/XHKgrpkp">neighborhood</a> is not necessarily open. In a topological space, one has the following facts:<a class="footnote-ref" id="fnref:38" href="#fn:38">38</a>

<ul><li>If 
 
 
 
 U
 
 
 {\displaystyle U}
 
 is a neighborhood of 
 
 
 
 x
 
 
 {\displaystyle x}
 
 then 
 
 
 
 x
 
 
 {\displaystyle x}
 
 is an element of 
 
 
 
 U
 .
 
 
 {\displaystyle U.}
 
</li>
<li>The intersection of two neighborhoods of 
 
 
 
 x
 
 
 {\displaystyle x}
 
 is a neighborhood of 
 
 
 
 x
 .
 
 
 {\displaystyle x.}
 
 Equivalently, the intersection of finitely many neighborhoods of 
 
 
 
 x
 
 
 {\displaystyle x}
 
 is a neighborhood of 
 
 
 
 x
 .
 
 
 {\displaystyle x.}
 
</li>
<li>If 
 
 
 
 V
 
 
 {\displaystyle V}
 
 contains a neighborhood of 
 
 
 
 x
 ,
 
 
 {\displaystyle x,}
 
 then 
 
 
 
 V
 
 
 {\displaystyle V}
 
 is a neighborhood of 
 
 
 
 x
 .
 
 
 {\displaystyle x.}
 
</li>
<li>If 
 
 
 
 U
 
 
 {\displaystyle U}
 
 is a neighborhood of 
 
 
 
 x
 ,
 
 
 {\displaystyle x,}
 
 then there exists a neighborhood 
 
 
 
 V
 
 
 {\displaystyle V}
 
 of 
 
 
 
 x
 
 
 {\displaystyle x}
 
 such that 
 
 
 
 U
 
 
 {\displaystyle U}
 
 is a neighborhood of each point of 
 
 
 
 V
 
 
 {\displaystyle V}
 
.</li></ul>
If 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is a set and one declares a nonempty collection of neighborhoods for every point of 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 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.<a class="footnote-ref" id="fnref:39" href="#fn:39">39</a> It follows that on a topological space 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 all definitions can be phrased in terms of neighborhoods:

<ul><li>Given another topological space 
 
 
 
 Y
 ,
 
 
 {\displaystyle Y,}
 
 a map 
 
 
 
 f
 :
 X
 →
 Y
 
 
 {\displaystyle f:X\to Y}
 
 is continuous if and only for every element 
 
 
 
 x
 
 
 {\displaystyle x}
 
 of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and every neighborhood 
 
 
 
 B
 
 
 {\displaystyle B}
 
 of 
 
 
 
 f
 (
 x
 )
 ,
 
 
 {\displaystyle f(x),}
 
 the preimage 
 
 
 
 
 f
 
 −
 1
 
 
 (
 B
 )
 
 
 {\displaystyle f^{-1}(B)}
 
 is a neighborhood of 
 
 
 
 x
 .
 
 
 {\displaystyle x.}
 
<a class="footnote-ref" id="fnref:40" href="#fn:40">40</a></li>
<li>A subset of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is open if and only if it is a neighborhood of each of its points.</li>
<li>Given a subset 
 
 
 
 A
 
 
 {\displaystyle A}
 
 of 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 the interior is the collection of all elements 
 
 
 
 x
 
 
 {\displaystyle x}
 
 of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 such that 
 
 
 
 A
 
 
 {\displaystyle A}
 
 is a neighbourhood of 
 
 
 
 x
 
 
 {\displaystyle x}
 
.</li>
<li>Given a subset 
 
 
 
 A
 
 
 {\displaystyle A}
 
 of 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 the closure is the collection of all elements 
 
 
 
 x
 
 
 {\displaystyle x}
 
 of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 such that every neighborhood of 
 
 
 
 x
 
 
 {\displaystyle x}
 
 intersects 
 
 
 
 A
 .
 
 
 {\displaystyle A.}
 
<a class="footnote-ref" id="fnref:41" href="#fn:41">41</a></li></ul>
<h2 id="definition-via-convergence-of-nets">Definition via convergence of nets</h2>
See also: <a href="/facts/Net_(mathematics)/msmsvMTT">Net (mathematics)</a> and <a href="/facts/Fr%25C3%25A9chet%25E2%2580%2593Urysohn_space/ccN00rSh">Fréchet–Urysohn space</a>
Convergence of nets satisfies the following properties:<a class="footnote-ref" id="fnref:42" href="#fn:42">42</a><a class="footnote-ref" id="fnref:43" href="#fn:43">43</a>

<ol><li>Every constant net converges to itself.</li>
<li>Every <a href="/facts/Subnet_(mathematics)/UWj0v083">subnet</a> of a convergent net converges to the same limits.</li>
<li>If a net does not converge to a point 
 
 
 
 x
 
 
 {\displaystyle x}
 
 then there is a subnet such that no further subnet converges to 
 
 
 
 x
 .
 
 
 {\displaystyle x.}
 
 Equivalently, if 
 
 
 
 
 x
 
 ∙
 
 
 
 
 {\displaystyle x_{\bullet }}
 
 is a net such that every one of its subnets has a sub-subnet that converges to a point 
 
 
 
 x
 ,
 
 
 {\displaystyle x,}
 
 then 
 
 
 
 
 x
 
 ∙
 
 
 
 
 {\displaystyle x_{\bullet }}
 
 converges to 
 
 
 
 x
 .
 
 
 {\displaystyle x.}
 
</li>
<li>Diagonal principle/Convergence of iterated limits. If 
 
 
 
 
 
 (
 
 x
 
 a
 
 
 )
 
 
 a
 ∈
 A
 
 
 →
 x
 
 
 {\displaystyle \left(x_{a}\right)_{a\in A}\to x}
 
 in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and for every index 
 
 
 
 a
 ∈
 A
 ,
 
 
 {\displaystyle a\in A,}
 
 
 
 
 
 
 
 (
 
 x
 
 a
 
 
 i
 
 
 )
 
 
 i
 ∈
 
 I
 
 a
 
 
 
 
 
 
 {\displaystyle \left(x_{a}^{i}\right)_{i\in I_{a}}}
 
 is a net that <a href="/facts/Convergent_net/msmsvMTT">converges to</a> 
 
 
 
 
 x
 
 a
 
 
 
 
 {\displaystyle x_{a}}
 
 in 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 then there exists a diagonal (sub)net of 
 
 
 
 
 
 (
 
 x
 
 a
 
 
 i
 
 
 )
 
 
 a
 ∈
 A
 ,
 i
 ∈
 
 I
 
 a
 
 
 
 
 
 
 {\displaystyle \left(x_{a}^{i}\right)_{a\in A,i\in I_{a}}}
 
 that converges to 
 
 
 
 x
 .
 
 
 {\displaystyle x.}
 
</li></ol>
<ul><li>A diagonal net refers to any <a href="/facts/Subnet_(mathematics)/UWj0v083">subnet</a> of 
 
 
 
 
 
 (
 
 x
 
 a
 
 
 i
 
 
 )
 
 
 a
 ∈
 A
 ,
 i
 ∈
 
 I
 
 a
 
 
 
 
 .
 
 
 {\displaystyle \left(x_{a}^{i}\right)_{a\in A,i\in I_{a}}.}
 
</li>
<li>The : notation 
 
 
 
 
 
 (
 
 x
 
 a
 
 
 i
 
 
 )
 
 
 a
 ∈
 A
 ,
 i
 ∈
 
 I
 
 a
 
 
 
 
 
 
 {\displaystyle \left(x_{a}^{i}\right)_{a\in A,i\in I_{a}}}
 
 denotes the net defined by 
 
 
 
 (
 a
 ,
 i
 )
 ↦
 
 x
 
 a
 
 
 i
 
 
 
 
 {\displaystyle (a,i)\mapsto x_{a}^{i}}
 
 whose domain is the set 
 
 
 
 
 
 
 ⋃
 
 a
 ∈
 A
 
 
 
 
 A
 ×
 
 I
 
 a
 
 
 
 
 {\displaystyle {\textstyle \bigcup \limits _{a\in A}}A\times I_{a}}
 
 <a href="/facts/Lexicographic_order/xQANy8Br">ordered lexicographically</a> first by 
 
 
 
 A
 
 
 {\displaystyle A}
 
 and then by 
 
 
 
 
 I
 
 a
 
 
 ;
 
 
 {\displaystyle I_{a};}
 
<a class="footnote-ref" id="fnref:44" href="#fn:44">44</a> explicitly, given any two pairs 
 
 
 
 (
 
 a
 
 1
 
 
 ,
 
 i
 
 1
 
 
 )
 ,
 
 (
 
 
 a
 
 2
 
 
 ,
 
 i
 
 2
 
 
 
 )
 
 ∈
 
 
 
 ⋃
 
 a
 ∈
 A
 
 
 
 
 A
 ×
 
 I
 
 a
 
 
 ,
 
 
 {\displaystyle (a_{1},i_{1}),\left(a_{2},i_{2}\right)\in {\textstyle \bigcup \limits _{a\in A}}A\times I_{a},}
 
 declare that 
 
 
 
 (
 
 a
 
 1
 
 
 ,
 
 i
 
 1
 
 
 )
 ≤
 
 (
 
 
 a
 
 2
 
 
 ,
 
 i
 
 2
 
 
 
 )
 
 
 
 {\displaystyle (a_{1},i_{1})\leq \left(a_{2},i_{2}\right)}
 
 holds if and only if both (1) 
 
 
 
 
 a
 
 1
 
 
 ≤
 
 a
 
 2
 
 
 ,
 
 
 {\displaystyle a_{1}\leq a_{2},}
 
 and also (2) if 
 
 
 
 
 a
 
 1
 
 
 =
 
 a
 
 2
 
 
 
 
 {\displaystyle a_{1}=a_{2}}
 
 then 
 
 
 
 
 i
 
 1
 
 
 ≤
 
 i
 
 2
 
 
 .
 
 
 {\displaystyle i_{1}\leq i_{2}.}
 
</li></ul>

If 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is a set, then given a notion of net convergence (telling what nets converge to what points<a class="footnote-ref" id="fnref:45" href="#fn:45">45</a>) satisfying the above four axioms, a closure operator on 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is defined by sending any given set 
 
 
 
 A
 
 
 {\displaystyle A}
 
 to the set of all limits of all nets valued in 
 
 
 
 A
 ;
 
 
 {\displaystyle A;}
 
 the corresponding topology is the unique topology inducing the given convergences of nets to points.<a class="footnote-ref" id="fnref:46" href="#fn:46">46</a> 
Given a subset 
 
 
 
 A
 ⊆
 X
 
 
 {\displaystyle A\subseteq X}
 
 of a topological space 
 
 
 
 X
 :
 
 
 {\displaystyle X:}

<ul><li>
 
 
 
 A
 
 
 {\displaystyle A}
 
 is open in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 if and only if every net converging to an element of 
 
 
 
 A
 
 
 {\displaystyle A}
 
 is eventually contained in 
 
 
 
 A
 .
 
 
 {\displaystyle A.}
 
</li>
<li>the closure of 
 
 
 
 A
 
 
 {\displaystyle A}
 
 in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is the set of all limits of all convergent nets valued in 
 
 
 
 A
 .
 
 
 {\displaystyle A.}
 
<a class="footnote-ref" id="fnref:47" href="#fn:47">47</a><a class="footnote-ref" id="fnref:48" href="#fn:48">48</a></li>
<li>
 
 
 
 A
 
 
 {\displaystyle A}
 
 is closed in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 if and only if there does not exist a net in 
 
 
 
 A
 
 
 {\displaystyle A}
 
 that converges to an element of the complement 
 
 
 
 X
 ∖
 A
 .
 
 
 {\displaystyle X\setminus A.}
 
<a class="footnote-ref" id="fnref:49" href="#fn:49">49</a> A subset 
 
 
 
 A
 ⊆
 X
 
 
 {\displaystyle A\subseteq X}
 
 is closed in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 if and only if every limit point of every convergent net in 
 
 
 
 A
 
 
 {\displaystyle A}
 
 necessarily belongs to 
 
 
 
 A
 .
 
 
 {\displaystyle A.}
 
<a class="footnote-ref" id="fnref:50" href="#fn:50">50</a></li></ul>
A function 
 
 
 
 f
 :
 X
 →
 Y
 
 
 {\displaystyle f:X\to Y}
 
 between two topological spaces is continuous if and only if for every 
 
 
 
 x
 ∈
 X
 
 
 {\displaystyle x\in X}
 
 and every net 
 
 
 
 
 x
 
 ∙
 
 
 
 
 {\displaystyle x_{\bullet }}
 
 in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 that <a href="/facts/Convergent_net/msmsvMTT">converges to</a> 
 
 
 
 x
 
 
 {\displaystyle x}
 
 in 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 the net 
 
 
 
 f
 
 (
 
 x
 
 ∙
 
 
 )
 
 
 
 {\displaystyle f\left(x_{\bullet }\right)}
 
<a class="footnote-ref" id="fnref:51" href="#fn:51">51</a> converges to 
 
 
 
 f
 (
 x
 )
 
 
 {\displaystyle f(x)}
 
 in 
 
 
 
 Y
 .
 
 
 {\displaystyle Y.}
 
<a class="footnote-ref" id="fnref:52" href="#fn:52">52</a>

<h2 id="definition-via-convergence-of-filters">Definition via convergence of filters</h2>
See also: <a href="/facts/Filters_in_topology/5vQbHcpE">Filters in topology</a>
A topology can also be defined on a set by declaring which filters converge to which points. One has the following characterizations of standard objects in terms of <a href="/facts/Filter_(set_theory)/7f3zk55u">filters</a> and <a href="/facts/Prefilter/7f3zk55u">prefilters</a> (also known as filterbases):

<ul><li>Given a second topological space 
 
 
 
 Y
 ,
 
 
 {\displaystyle Y,}
 
 a function 
 
 
 
 f
 :
 X
 →
 Y
 
 
 {\displaystyle f:X\to Y}
 
 is continuous if and only if it preserves <a href="/facts/Convergent_prefilter/5vQbHcpE">convergence of prefilters</a>.<a class="footnote-ref" id="fnref:53" href="#fn:53">53</a></li>
<li>A subset 
 
 
 
 A
 
 
 {\displaystyle A}
 
 of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is open if and only if every <a href="/facts/Convergent_filter/5vQbHcpE">filter converging</a> to an element of 
 
 
 
 A
 
 
 {\displaystyle A}
 
 contains 
 
 
 
 A
 .
 
 
 {\displaystyle A.}
 
<a class="footnote-ref" id="fnref:54" href="#fn:54">54</a></li>
<li>A subset 
 
 
 
 A
 
 
 {\displaystyle A}
 
 of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is closed if and only if there does not exist a prefilter on 
 
 
 
 A
 
 
 {\displaystyle A}
 
 which converges to a point in the complement 
 
 
 
 X
 ∖
 A
 .
 
 
 {\displaystyle X\setminus A.}
 
<a class="footnote-ref" id="fnref:55" href="#fn:55">55</a></li>
<li>Given a subset 
 
 
 
 A
 
 
 {\displaystyle A}
 
 of 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 the closure consists of all points 
 
 
 
 x
 
 
 {\displaystyle x}
 
 for which there is a prefilter on 
 
 
 
 A
 
 
 {\displaystyle A}
 
 converging to 
 
 
 
 x
 .
 
 
 {\displaystyle x.}
 
<a class="footnote-ref" id="fnref:56" href="#fn:56">56</a></li>
<li>A subset 
 
 
 
 A
 
 
 {\displaystyle A}
 
 of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is a neighborhood of 
 
 
 
 x
 
 
 {\displaystyle x}
 
 if and only if it is an element of every filter converging to 
 
 
 
 x
 .
 
 
 {\displaystyle x.}
 
<a class="footnote-ref" id="fnref:57" href="#fn:57">57</a></li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Cauchy_space/1Q8kEebw">Cauchy space</a> – Concept in general topology and analysis</li>
<li><a href="/facts/Convergence_space/FJcNDeog">Convergence space</a> – Generalization of the notion of convergence that is found in general topology</li>
<li><a href="/facts/Filters_in_topology/5vQbHcpE">Filters in topology</a> – Use of filters to describe and characterize all basic topological notions and results.</li>
<li><a href="/facts/Sequential_space/xAjFpdfR">Sequential space</a> – Topological space characterized by sequences</li>
<li><a href="/facts/Topology_(structure)/v2ut7CbK">Topology (structure)</a> – Collection of open subsets of a topological space</li></ul>
<h2 id="citations">Citations</h2>

Notes

<ul><li><a href="/facts/James_Dugundji/xCy4DgGy">Dugundji, James</a> (1978). Topology. Allyn and Bacon Series in Advanced Mathematics (Reprinting of the 1966 original ed.). Boston, Mass.–London–Sydney: Allyn and Bacon, Inc.</li>
<li><a href="/facts/Ryszard_Engelking/aAjuMWvC">Engelking, Ryszard</a> (1977). General topology. Monografie Matematyczne. Vol. 60 (Translated by author from Polish ed.). Warsaw: PWN—Polish Scientific Publishers.</li>
<li><a href="/facts/John_L._Kelley/zmQyF2xn">Kelley, John L.</a> (1975). General topology. Graduate Texts in Mathematics. Vol. 27 (Reprint of the 1955 ed.). New York-Berlin: Springer-Verlag.</li>
<li><a href="/facts/Kazimierz_Kuratowski/8xoKvo5b">Kuratowski, K.</a> (1966). Topology. Vol. I. (Translated from the French by J. Jaworowski. Revised and augmented ed.). New York-London/Warsaw: Academic Press/Państwowe Wydawnictwo Naukowe.</li>
<li>Willard, Stephen (2004) [1970]. <a href="https://books.google.com/books?id=-o8xJQ7Ag2cC">General Topology</a>. <a href="/facts/Mineola%2c_N.Y./GWYvEokQ">Mineola, N.Y.</a>: <a href="/facts/Dover_Publications/RwjvlSFv">Dover Publications</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-486-43479-7. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/115240">115240</a>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Dugundji 1966; Engelking 1977; Kelley 1955. - Dugundji, James (1978). Topology. Allyn and Bacon Series in Advanced Mathematics (Reprinting of the 1966 original ed.). Boston, Mass.–London–Sydney: Allyn and Bacon, Inc. <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Kuratowski 1966, p.38. - Kuratowski, K. (1966). Topology. Vol. I. (Translated from the French by J. Jaworowski. Revised and augmented ed.). New York-London/Warsaw: Academic Press/Państwowe Wydawnictwo Naukowe. <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Dugundji 1966, p.62; Engelking 1977, p.11-12; Kelley 1955, p.37; Kuratowski 1966, p.45. - Dugundji, James (1978). Topology. Allyn and Bacon Series in Advanced Mathematics (Reprinting of the 1966 original ed.). Boston, Mass.–London–Sydney: Allyn and Bacon, Inc. <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Dugundji 1966, p.79; Engelking 1977, p.27-28; Kelley 1955, p.85; Kuratowski 1966, p.105. - Dugundji, James (1978). Topology. Allyn and Bacon Series in Advanced Mathematics (Reprinting of the 1966 original ed.). Boston, Mass.–London–Sydney: Allyn and Bacon, Inc. <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Dugundji 1966, p.68; Engelking 1977, p.13; Kelley 1955, p.40. - Dugundji, James (1978). Topology. Allyn and Bacon Series in Advanced Mathematics (Reprinting of the 1966 original ed.). Boston, Mass.–London–Sydney: Allyn and Bacon, Inc. <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Dugundji 1966, p.69; Engelking 1977, p.13. - Dugundji, James (1978). Topology. Allyn and Bacon Series in Advanced Mathematics (Reprinting of the 1966 original ed.). Boston, Mass.–London–Sydney: Allyn and Bacon, Inc. <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Dugundji 1966, p.71; Engelking 1977, p.14; Kelley 1955, p.44; Kuratowski 1966, p.58. - Dugundji, James (1978). Topology. Allyn and Bacon Series in Advanced Mathematics (Reprinting of the 1966 original ed.). Boston, Mass.–London–Sydney: Allyn and Bacon, Inc. <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Kelley 1955, p.38; Kuratowski 1966, p.61. - Kelley, John L. (1975). General topology. Graduate Texts in Mathematics. Vol. 27 (Reprint of the 1955 ed.). New York-Berlin: Springer-Verlag. <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Dugundji 1966, p.63; Engelking 1977, p.12. - Dugundji, James (1978). Topology. Allyn and Bacon Series in Advanced Mathematics (Reprinting of the 1966 original ed.). Boston, Mass.–London–Sydney: Allyn and Bacon, Inc. <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Dugundji 1966, p.210; Engelking 1977, p.49; Kelley 1955, p.66; Kuratowski 1966, p.203. - Dugundji, James (1978). Topology. Allyn and Bacon Series in Advanced Mathematics (Reprinting of the 1966 original ed.). Boston, Mass.–London–Sydney: Allyn and Bacon, Inc. <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Engelking 1977, p.52; Kelley 1955, p.83. - Engelking, Ryszard (1977). General topology. Monografie Matematyczne. Vol. 60 (Translated by author from Polish ed.). Warsaw: PWN—Polish Scientific Publishers. <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Kuratowski 1966, p.6. - Kuratowski, K. (1966). Topology. Vol. I. (Translated from the French by J. Jaworowski. Revised and augmented ed.). New York-London/Warsaw: Academic Press/Państwowe Wydawnictwo Naukowe. <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Engelking 1977, p.52; Kelley 1955, p.83; Kuratowski 1966, p.63. - Engelking, Ryszard (1977). General topology. Monografie Matematyczne. Vol. 60 (Translated by author from Polish ed.). Warsaw: PWN—Polish Scientific Publishers. <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Dugundji 1966, 211; Engelking 1977, p.52. - Dugundji, James (1978). Topology. Allyn and Bacon Series in Advanced Mathematics (Reprinting of the 1966 original ed.). Boston, Mass.–London–Sydney: Allyn and Bacon, Inc. <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">Dugundji 1966, p.212; Engelking 1977, p.52. - Dugundji, James (1978). Topology. Allyn and Bacon Series in Advanced Mathematics (Reprinting of the 1966 original ed.). Boston, Mass.–London–Sydney: Allyn and Bacon, Inc. <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">Dugundji 1966, p.69; Engelking 1977, p.13; Kelley 1955, p.40; Kuratowski 1966, p.44. - Dugundji, James (1978). Topology. Allyn and Bacon Series in Advanced Mathematics (Reprinting of the 1966 original ed.). Boston, Mass.–London–Sydney: Allyn and Bacon, Inc. <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
<li id="fn:17">Dugundji 1966, p.74; Engelking 1977, p.22; Kelley 1955, p.40; Kuratowski 1966, p.44. - Dugundji, James (1978). Topology. Allyn and Bacon Series in Advanced Mathematics (Reprinting of the 1966 original ed.). Boston, Mass.–London–Sydney: Allyn and Bacon, Inc. <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
<li id="fn:18">Dugundji 1966, p.79; Engelking 1977, p.28; Kelley 1955, p.86; Kuratowski 1966, p.105. - Dugundji, James (1978). Topology. Allyn and Bacon Series in Advanced Mathematics (Reprinting of the 1966 original ed.). Boston, Mass.–London–Sydney: Allyn and Bacon, Inc. <a href="#fnref:18" class="footnote-back-ref">↩</a></li>
<li id="fn:19">Kelley 1955, p.41. - Kelley, John L. (1975). General topology. Graduate Texts in Mathematics. Vol. 27 (Reprint of the 1955 ed.). New York-Berlin: Springer-Verlag. <a href="#fnref:19" class="footnote-back-ref">↩</a></li>
<li id="fn:20">Dugundji 1966, p.70; Engelking 1977; Kelley 1955, p.42. - Dugundji, James (1978). Topology. Allyn and Bacon Series in Advanced Mathematics (Reprinting of the 1966 original ed.). Boston, Mass.–London–Sydney: Allyn and Bacon, Inc. <a href="#fnref:20" class="footnote-back-ref">↩</a></li>
<li id="fn:21">Dugundji 1966, p.69-70; Engelking 1977, p.14; Kelley 1955, p.42-43. - Dugundji, James (1978). Topology. Allyn and Bacon Series in Advanced Mathematics (Reprinting of the 1966 original ed.). Boston, Mass.–London–Sydney: Allyn and Bacon, Inc. <a href="#fnref:21" class="footnote-back-ref">↩</a></li>
<li id="fn:22">Dugundji 1966, p.73; Engelking 1977, p.22; Kelley 1955, p.43. - Dugundji, James (1978). Topology. Allyn and Bacon Series in Advanced Mathematics (Reprinting of the 1966 original ed.). Boston, Mass.–London–Sydney: Allyn and Bacon, Inc. <a href="#fnref:22" class="footnote-back-ref">↩</a></li>
<li id="fn:23">Dugundji 1966, p.80; Engelking 1977, p.28; Kelley 1955, p.86; Kuratowski 1966, p.105. - Dugundji, James (1978). Topology. Allyn and Bacon Series in Advanced Mathematics (Reprinting of the 1966 original ed.). Boston, Mass.–London–Sydney: Allyn and Bacon, Inc. <a href="#fnref:23" class="footnote-back-ref">↩</a></li>
<li id="fn:24">Kuratowski 1966, p.43. - Kuratowski, K. (1966). Topology. Vol. I. (Translated from the French by J. Jaworowski. Revised and augmented ed.). New York-London/Warsaw: Academic Press/Państwowe Wydawnictwo Naukowe. <a href="#fnref:24" class="footnote-back-ref">↩</a></li>
<li id="fn:25">Dugundji 1966, p.69; Kelley 1955, p.42; Kuratowski 1966, p.43. - Dugundji, James (1978). Topology. Allyn and Bacon Series in Advanced Mathematics (Reprinting of the 1966 original ed.). Boston, Mass.–London–Sydney: Allyn and Bacon, Inc. <a href="#fnref:25" class="footnote-back-ref">↩</a></li>
<li id="fn:26">Dugundji 1966, p.71; Engelking 1977, p.15; Kelley 1955, p.44-45; Kuratowski 1966, p.55. - Dugundji, James (1978). Topology. Allyn and Bacon Series in Advanced Mathematics (Reprinting of the 1966 original ed.). Boston, Mass.–London–Sydney: Allyn and Bacon, Inc. <a href="#fnref:26" class="footnote-back-ref">↩</a></li>
<li id="fn:27">Engelking 1977, p.15. - Engelking, Ryszard (1977). General topology. Monografie Matematyczne. Vol. 60 (Translated by author from Polish ed.). Warsaw: PWN—Polish Scientific Publishers. <a href="#fnref:27" class="footnote-back-ref">↩</a></li>
<li id="fn:28">Dugundji 1966, p.74; Engelking 1977, p.23. - Dugundji, James (1978). Topology. Allyn and Bacon Series in Advanced Mathematics (Reprinting of the 1966 original ed.). Boston, Mass.–London–Sydney: Allyn and Bacon, Inc. <a href="#fnref:28" class="footnote-back-ref">↩</a></li>
<li id="fn:29">Engelking 1977, p.28; Kuratowski 1966, p.103. - Engelking, Ryszard (1977). General topology. Monografie Matematyczne. Vol. 60 (Translated by author from Polish ed.). Warsaw: PWN—Polish Scientific Publishers. <a href="#fnref:29" class="footnote-back-ref">↩</a></li>
<li id="fn:30">Dugundji 1966, p.71; Kelley 1955, p.44. - Dugundji, James (1978). Topology. Allyn and Bacon Series in Advanced Mathematics (Reprinting of the 1966 original ed.). Boston, Mass.–London–Sydney: Allyn and Bacon, Inc. <a href="#fnref:30" class="footnote-back-ref">↩</a></li>
<li id="fn:31">Kelley 1955, p.44-45. - Kelley, John L. (1975). General topology. Graduate Texts in Mathematics. Vol. 27 (Reprint of the 1955 ed.). New York-Berlin: Springer-Verlag. <a href="#fnref:31" class="footnote-back-ref">↩</a></li>
<li id="fn:32">Lei, Yinbin; Zhang, Jun (August 2019). "Generalizing Topological Set Operators". Electronic Notes in Theoretical Computer Science. 345: 63–76. doi:10.1016/j.entcs.2019.07.016. ISSN 1571-0661. <a href="https://doi.org/10.1016%2Fj.entcs.2019.07.016" target="_blank">https://doi.org/10.1016%2Fj.entcs.2019.07.016</a> <a href="#fnref:32" class="footnote-back-ref">↩</a></li>
<li id="fn:33">Bourbaki, Nicolas (1998). Elements of mathematics. Chapters 1/4: 3. General topology Chapters 1 - 4 (Softcover ed., [Nachdr.] - [1998] ed.). Berlin Heidelberg: Springer. ISBN 978-3-540-64241-1. <a href="978-3-540-64241-1" target="_blank">978-3-540-64241-1</a> <a href="#fnref:33" class="footnote-back-ref">↩</a></li>
<li id="fn:34">Lei, Yinbin; Zhang, Jun (August 2019). "Generalizing Topological Set Operators". Electronic Notes in Theoretical Computer Science. 345: 63–76. doi:10.1016/j.entcs.2019.07.016. ISSN 1571-0661. <a href="https://doi.org/10.1016%2Fj.entcs.2019.07.016" target="_blank">https://doi.org/10.1016%2Fj.entcs.2019.07.016</a> <a href="#fnref:34" class="footnote-back-ref">↩</a></li>
<li id="fn:35">Lei, Yinbin; Zhang, Jun (August 2019). "Generalizing Topological Set Operators". Electronic Notes in Theoretical Computer Science. 345: 63–76. doi:10.1016/j.entcs.2019.07.016. ISSN 1571-0661. <a href="https://doi.org/10.1016%2Fj.entcs.2019.07.016" target="_blank">https://doi.org/10.1016%2Fj.entcs.2019.07.016</a> <a href="#fnref:35" class="footnote-back-ref">↩</a></li>
<li id="fn:36">Baker, Crump W. (1991). Introduction to topology. Dubuque, IA: Wm. C. Brown Publishers. ISBN 978-0-697-05972-7. <a href="978-0-697-05972-7" target="_blank">978-0-697-05972-7</a> <a href="#fnref:36" class="footnote-back-ref">↩</a></li>
<li id="fn:37">Hocking, John G.; Young, Gail S. (1988). Topology. New York: Dover Publications. ISBN 978-0-486-65676-2. <a href="978-0-486-65676-2" target="_blank">978-0-486-65676-2</a> <a href="#fnref:37" class="footnote-back-ref">↩</a></li>
<li id="fn:38">Willard 2004, pp. 31–32. - Willard, Stephen (2004) [1970]. General Topology. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-43479-7. OCLC 115240. <a href="https://books.google.com/books?id=-o8xJQ7Ag2cC" target="_blank">https://books.google.com/books?id=-o8xJQ7Ag2cC</a> <a href="#fnref:38" class="footnote-back-ref">↩</a></li>
<li id="fn:39">Willard 2004, pp. 31–32. - Willard, Stephen (2004) [1970]. General Topology. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-43479-7. OCLC 115240. <a href="https://books.google.com/books?id=-o8xJQ7Ag2cC" target="_blank">https://books.google.com/books?id=-o8xJQ7Ag2cC</a> <a href="#fnref:39" class="footnote-back-ref">↩</a></li>
<li id="fn:40">Kuratowski 1966, p.103. - Kuratowski, K. (1966). Topology. Vol. I. (Translated from the French by J. Jaworowski. Revised and augmented ed.). New York-London/Warsaw: Academic Press/Państwowe Wydawnictwo Naukowe. <a href="#fnref:40" class="footnote-back-ref">↩</a></li>
<li id="fn:41">Kuratowski 1966, p.61. - Kuratowski, K. (1966). Topology. Vol. I. (Translated from the French by J. Jaworowski. Revised and augmented ed.). New York-London/Warsaw: Academic Press/Państwowe Wydawnictwo Naukowe. <a href="#fnref:41" class="footnote-back-ref">↩</a></li>
<li id="fn:42">Kelley 1955, p.74. - Kelley, John L. (1975). General topology. Graduate Texts in Mathematics. Vol. 27 (Reprint of the 1955 ed.). New York-Berlin: Springer-Verlag. <a href="#fnref:42" class="footnote-back-ref">↩</a></li>
<li id="fn:43">Willard 2004, p. 77. - Willard, Stephen (2004) [1970]. General Topology. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-43479-7. OCLC 115240. <a href="https://books.google.com/books?id=-o8xJQ7Ag2cC" target="_blank">https://books.google.com/books?id=-o8xJQ7Ag2cC</a> <a href="#fnref:43" class="footnote-back-ref">↩</a></li>
<li id="fn:44">Willard 2004, p. 77. - Willard, Stephen (2004) [1970]. General Topology. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-43479-7. OCLC 115240. <a href="https://books.google.com/books?id=-o8xJQ7Ag2cC" target="_blank">https://books.google.com/books?id=-o8xJQ7Ag2cC</a> <a href="#fnref:44" class="footnote-back-ref">↩</a></li>
<li id="fn:45">Willard 2004, p. 77. - Willard, Stephen (2004) [1970]. General Topology. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-43479-7. OCLC 115240. <a href="https://books.google.com/books?id=-o8xJQ7Ag2cC" target="_blank">https://books.google.com/books?id=-o8xJQ7Ag2cC</a> <a href="#fnref:45" class="footnote-back-ref">↩</a></li>
<li id="fn:46">Kelley 1955, p.74. - Kelley, John L. (1975). General topology. Graduate Texts in Mathematics. Vol. 27 (Reprint of the 1955 ed.). New York-Berlin: Springer-Verlag. <a href="#fnref:46" class="footnote-back-ref">↩</a></li>
<li id="fn:47">Engelking 1977, p.50; Kelley 1955, p.66. - Engelking, Ryszard (1977). General topology. Monografie Matematyczne. Vol. 60 (Translated by author from Polish ed.). Warsaw: PWN—Polish Scientific Publishers. <a href="#fnref:47" class="footnote-back-ref">↩</a></li>
<li id="fn:48">Willard 2004, p. 77. - Willard, Stephen (2004) [1970]. General Topology. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-43479-7. OCLC 115240. <a href="https://books.google.com/books?id=-o8xJQ7Ag2cC" target="_blank">https://books.google.com/books?id=-o8xJQ7Ag2cC</a> <a href="#fnref:48" class="footnote-back-ref">↩</a></li>
<li id="fn:49">Engelking 1977, p.51; Kelley 1955, p.66. - Engelking, Ryszard (1977). General topology. Monografie Matematyczne. Vol. 60 (Translated by author from Polish ed.). Warsaw: PWN—Polish Scientific Publishers. <a href="#fnref:49" class="footnote-back-ref">↩</a></li>
<li id="fn:50">Willard 2004, pp. 73–77. - Willard, Stephen (2004) [1970]. General Topology. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-43479-7. OCLC 115240. <a href="https://books.google.com/books?id=-o8xJQ7Ag2cC" target="_blank">https://books.google.com/books?id=-o8xJQ7Ag2cC</a> <a href="#fnref:50" class="footnote-back-ref">↩</a></li>
<li id="fn:51">Assuming that the net 
 
 
 
 
 x
 
 ∙
 
 
 
 
 {\displaystyle x_{\bullet }}
 
 is indexed by 
 
 
 
 I
 
 
 {\displaystyle I}
 
 (so that 
 
 
 
 
 x
 
 ∙
 
 
 =
 
 
 (
 
 x
 
 i
 
 
 )
 
 
 i
 ∈
 I
 
 
 ,
 
 
 {\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I},}
 
 which is just notation for function 
 
 
 
 
 x
 
 ∙
 
 
 :
 I
 →
 X
 
 
 {\displaystyle x_{\bullet }:I\to X}
 
 that sends 
 
 
 
 i
 ↦
 
 x
 
 i
 
 
 
 
 {\displaystyle i\mapsto x_{i}}
 
) then 
 
 
 
 f
 
 (
 
 x
 
 ∙
 
 
 )
 
 
 
 {\displaystyle f\left(x_{\bullet }\right)}
 
 denotes the composition of 
 
 
 
 
 x
 
 ∙
 
 
 :
 I
 →
 X
 
 
 {\displaystyle x_{\bullet }:I\to X}
 
 with 
 
 
 
 f
 :
 X
 →
 Y
 .
 
 
 {\displaystyle f:X\to Y.}
 
 That is, 
 
 
 
 f
 
 (
 
 x
 
 ∙
 
 
 )
 
 :=
 f
 ∘
 
 x
 
 ∙
 
 
 =
 
 
 (
 
 f
 
 (
 
 x
 
 i
 
 
 )
 
 
 )
 
 
 i
 ∈
 I
 
 
 
 
 {\displaystyle f\left(x_{\bullet }\right):=f\circ x_{\bullet }=\left(f\left(x_{i}\right)\right)_{i\in I}}
 
 is the function 
 
 
 
 f
 ∘
 
 x
 
 ∙
 
 
 :
 I
 →
 Y
 .
 
 
 {\displaystyle f\circ x_{\bullet }:I\to Y.}
 <a href="/wiki/Function_composition" target="_blank">/wiki/Function_composition</a> <a href="#fnref:51" class="footnote-back-ref">↩</a></li>
<li id="fn:52">Engelking 1977, p.51; Kelley 1955, p.86. - Engelking, Ryszard (1977). General topology. Monografie Matematyczne. Vol. 60 (Translated by author from Polish ed.). Warsaw: PWN—Polish Scientific Publishers. <a href="#fnref:52" class="footnote-back-ref">↩</a></li>
<li id="fn:53">Dugundji 1966, p.216; Engelking 1977, p.52. - Dugundji, James (1978). Topology. Allyn and Bacon Series in Advanced Mathematics (Reprinting of the 1966 original ed.). Boston, Mass.–London–Sydney: Allyn and Bacon, Inc. <a href="#fnref:53" class="footnote-back-ref">↩</a></li>
<li id="fn:54">Kelley 1955, p.83. - Kelley, John L. (1975). General topology. Graduate Texts in Mathematics. Vol. 27 (Reprint of the 1955 ed.). New York-Berlin: Springer-Verlag. <a href="#fnref:54" class="footnote-back-ref">↩</a></li>
<li id="fn:55">Dugundji 1966, p.215. - Dugundji, James (1978). Topology. Allyn and Bacon Series in Advanced Mathematics (Reprinting of the 1966 original ed.). Boston, Mass.–London–Sydney: Allyn and Bacon, Inc. <a href="#fnref:55" class="footnote-back-ref">↩</a></li>
<li id="fn:56">Dugundji 1966, p.215; Engelking 1977, p.52. - Dugundji, James (1978). Topology. Allyn and Bacon Series in Advanced Mathematics (Reprinting of the 1966 original ed.). Boston, Mass.–London–Sydney: Allyn and Bacon, Inc. <a href="#fnref:56" class="footnote-back-ref">↩</a></li>
<li id="fn:57">Kelley 1955, p.83. - Kelley, John L. (1975). General topology. Graduate Texts in Mathematics. Vol. 27 (Reprint of the 1955 ed.). New York-Berlin: Springer-Verlag. <a href="#fnref:57" class="footnote-back-ref">↩</a></li>
</ol>

Axiomatic foundations of topological spaces open-in-new

Axiomatic foundations of topological spaces