Editing Open and closed maps (section)

==Definitions and characterizations==

If <math>S</math> is a subset of a topological space then let <math>\overline{S}</math> and <math>\operatorname{Cl} S</math> (resp. <math>\operatorname{Int} S</math>) denote the [[Closure (topology)|closure]] (resp. [[Interior (topology)|interior]]) of <math>S</math> in that space. 
Let <math>f : X \to Y</math> be a function between [[topological space]]s. If <math>S</math> is any set then <math>f(S) := \left\{ f(s) ~:~ s \in S \cap \operatorname{domain} f \right\}</math> is called the image of <math>S</math> under <math>f.</math> 

===Competing definitions===

There are two different competing, but closely related, definitions of "{{em|open map}}" that are widely used, where both of these definitions can be summarized as: "it is a map that sends open sets to open sets." 
The following terminology is sometimes used to distinguish between the two definitions. 

A map <math>f : X \to Y</math> is called a
* "'''{{em|Strongly open map}}'''" if whenever <math>U</math> is an [[Open set|open subset]] of the domain <math>X</math> then <math>f(U)</math> is an open subset of <math>f</math>'s [[codomain]] <math>Y.</math> 
* "'''{{em|{{visible anchor|Relatively open map}}}}'''" if whenever <math>U</math> is an open subset of the domain <math>X</math> then <math>f(U)</math> is an open 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>{{sfn|Narici|Beckenstein|2011|pp=225-273}} 

Every strongly open map is a relatively open map. However, these definitions are not equivalent in general.  

:'''Warning''': Many authors define "open map" to mean "{{em|relatively}} open map" (for example, The Encyclopedia of Mathematics) while others define "open map" to mean "{{em|strongly}} open map". In general, these definitions are {{em|not}} equivalent so it is thus advisable to always check what definition of "open map" an author is using. 

A [[Surjective function|surjective]] map is relatively open if and only if it is strongly open; so for this important special case the definitions are equivalent. 
More generally, a map <math>f : X \to Y</math> is relatively open if and only if the [[Surjective function|surjection]] <math>f : X \to f(X)</math> is a strongly open map. 

Because <math>X</math> is always an open subset of <math>X,</math> the image <math>f(X) = \operatorname{Im} f</math> of a strongly open map <math>f : X \to Y</math> must be an open subset of its codomain <math>Y.</math> In fact, a relatively open map is a strongly open map if and only if its image is an open subset of its codomain. 
In summary,

:A map is strongly open if and only if it is relatively open and its image is an open subset of its codomain.

By using this characterization, it is often straightforward to apply results involving one of these two definitions of "open map" to a situation involving the other definition. 

The discussion above will also apply to closed maps if each instance of the word "open" is replaced with the word "closed".

===Open maps===

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

<ol>
<li>Definition: <math>f : X \to Y</math> maps open subsets of its domain to open subsets of its codomain; that is, for any open subset <math>U</math> of <math>X</math>, <math>f(U)</math> is an open subset of <math>Y.</math></li>
<li><math>f : X \to Y</math> is a relatively open map and its image <math>\operatorname{Im} f := f(X)</math> is an open subset of its codomain <math>Y.</math></li>
<li>For every <math>x \in X</math> and every [[Neighborhood (topology)|neighborhood]] <math>N</math> of <math>x</math> (however small), <math>f(N)</math> is a neighborhood of <math>f(x)</math>. We can replace the first or both instances of the word "neighborhood" with "open neighborhood" in this condition and the result will still be an equivalent condition:
* For every <math>x \in X</math> and every open neighborhood <math>N</math> of <math>x</math>, <math>f(N)</math> is a neighborhood of <math>f(x)</math>.
* For every <math>x \in X</math> and every open neighborhood <math>N</math> of <math>x</math>, <math>f(N)</math> is an open neighborhood of <math>f(x)</math>.</li>
<li><math>f\left( \operatorname{Int}_X A \right) \subseteq \operatorname{Int}_Y ( f(A) )</math> for all subsets <math>A</math> of <math>X,</math> where <math>\operatorname{Int}</math> denotes the [[topological interior]] of the set.</li>
<li>Whenever <math>C</math> is a [[Closed set|closed subset]] of <math>X</math> then the set <math>\left\{ y \in Y ~:~ f^{-1}(y) \subseteq C \right\}</math> is a closed subset of <math>Y.</math>
* This is a consequence of the [[List of set identities and relations|identity]] <math>f(X \setminus R) = Y \setminus \left\{ y \in Y : f^{-1}(y) \subseteq R \right\},</math> which holds for all subsets <math>R \subseteq X.</math></li>
</ol>

If <math>\mathcal{B}</math> is a [[Base (topology)|basis]] for <math>X</math> then the following can be appended to this list:

# <li value="6"><math>f</math> maps basic open sets to open sets in its codomain (that is, for any basic open set <math>B \in \mathcal{B},</math> <math>f(B)</math> is an open subset of <math>Y</math>).</li>

===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.