Editing Open and closed maps (section)

===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>