Editing Open and closed maps (section)

===Closed maps===

A map <math>f : X \to Y</math> is called a '''{{em|{{visible anchor|relatively closed map}}}}''' if whenever <math>C</math> is a [[Closed set|closed subset]] of the domain <math>X</math> then <math>f(C)</math> is a closed subset of <math>f</math>'s [[Image (mathematics)|image]] <math>\operatorname{Im} f := f(X),</math> where as usual, this set is endowed with the [[subspace topology]] induced on it by <math>f</math>'s [[codomain]] <math>Y.</math> 

A map <math>f : X \to Y</math> is called a '''{{em|{{visible anchor|closed map}}}}''' or a '''{{em|{{visible anchor|strongly closed map}}}}''' if it satisfies any of the following equivalent conditions: 

<ol>
<li>Definition: <math>f : X \to Y</math> maps closed subsets of its domain to closed subsets of its codomain; that is, for any closed subset <math>C</math> of <math>X,</math> <math>f(C)</math> is a closed subset of <math>Y.</math>
<li><math>f : X \to Y</math> is a relatively closed map and its image <math>\operatorname{Im} f := f(X)</math> is a closed subset of its codomain <math>Y.</math></li>
<li><math>\overline{f(A)} \subseteq f\left(\overline{A}\right)</math> for every subset <math>A \subseteq X.</math></li>
<li><math>\overline{f(C)} \subseteq f(C)</math> for every closed subset <math>C \subseteq X.</math></li>
<li><math>\overline{f(C)} = f(C)</math> for every closed subset <math>C \subseteq X.</math></li>
<li>Whenever <math>U</math> is an open subset of <math>X</math> then the set <math>\left\{y \in Y ~:~ f^{-1}(y) \subseteq U\right\}</math> is an open subset of <math>Y.</math></li>
<li>If <math>x_{\bull}</math> is a [[Net (mathematics)|net]] in <math>X</math> and <math>y \in Y</math> is a point such that <math>f\left(x_{\bull}\right) \to y</math> in <math>Y,</math> then <math>x_{\bull}</math> converges in <math>X</math> to the set <math>f^{-1}(y).</math> 
* The convergence <math>x_{\bull} \to f^{-1}(y)</math> means that every open subset of <math>X</math> that contains <math>f^{-1}(y)</math> will contain <math>x_j</math> for all sufficiently large indices <math>j.</math></li>
</ol>

A [[Surjective function|surjective]] map is strongly closed if and only if it is relatively closed. So for this important special case, the two definitions are equivalent. 
By definition, the map <math>f : X \to Y</math> is a relatively closed map if and only if the [[Surjective function|surjection]] <math>f : X \to \operatorname{Im} f</math> is a strongly closed map. 

If in the open set definition of "[[Continuous function|continuous map]]" (which is the statement: "every preimage of an open set is open"), both instances of the word "open" are replaced with "closed" then the statement of results ("every preimage of a closed set is closed") is {{em|[[Logical equivalence|equivalent]]}} to continuity. 
This does not happen with the definition of "open map" (which is: "every image of an open set is open") since the statement that results ("every image of a closed set is closed") is the definition of "closed map", which is in general {{em|not}} equivalent to openness. There exist open maps that are not closed and there also exist closed maps that are not open. This difference between open/closed maps and continuous maps is ultimately due to the fact that for any set <math>S,</math> only <math>f(X \setminus S) \supseteq f(X) \setminus f(S)</math> is guaranteed in general, whereas for preimages, equality <math>f^{-1}(Y \setminus S) = f^{-1}(Y) \setminus f^{-1}(S)</math> always holds.