Editing Open and closed maps (section)

==Properties==

===Open or closed maps that are continuous===

If <math>f : X \to Y</math> is a continuous map that is also open {{em|or}} closed then: 
* if <math>f</math> is a surjection then it is a [[quotient map (topology)|quotient map]] and even a [[hereditarily quotient map]],
** A surjective map <math>f : X \to Y</math> is called {{em|hereditarily quotient}} if for every subset <math>T \subseteq Y,</math> the restriction <math>f\big\vert_{f^{-1}(T)} ~:~ f^{-1}(T) \to T</math> is a quotient map. 
* if <math>f</math> is an [[Injective function|injection]] then it is a [[topological embedding]].
* if <math>f</math> is a [[bijection]] then it is a [[homeomorphism]].

In the first two cases, being open or closed is merely a [[sufficient condition]] for the conclusion that follows. 
In the third case, it is [[Necessary condition|necessary]] as well. 

===Open continuous maps===

If <math>f : X \to Y</math> is a continuous (strongly) open map, <math>A \subseteq X,</math> and <math>S \subseteq Y,</math> then:

<ul>
<li><math>f^{-1}\left(\operatorname{Bd}_Y S\right) = \operatorname{Bd}_X \left(f^{-1}(S)\right)</math> where <math>\operatorname{Bd}</math> denotes the [[Boundary (topology)|boundary]] of a set.</li>
<li><math>f^{-1}\left(\overline{S}\right) = \overline{f^{-1}(S)}</math> where <math>\overline{S}</math> denote the [[Closure (topology)|closure]] of a set.</li>
<li>If <math>\overline{A} = \overline{\operatorname{Int}_X A},</math> where <math>\operatorname{Int} </math> denotes the [[Interior (topology)|interior]] of a set, then 
<math display="block">\overline{\operatorname{Int}_Y f(A)} = \overline{f(A)} = \overline{f\left(\operatorname{Int}_X A\right)} = \overline{f \left(\overline{\operatorname{Int}_X A}\right)}</math>
where this set <math>\overline{f(A)}</math> is also necessarily a [[regular closed set]] (in <math>Y</math>).<ref group=note name="DefOfRegularOpenClosed" /> In particular, if <math>A</math> is a regular closed set then so is <math>\overline{f(A)}.</math> And if <math>A</math> is a [[regular open set]] then so is <math>Y \setminus \overline{f(X \setminus A)}.</math> 
</li>
<li>If the continuous open map <math>f : X \to Y</math> is also surjective then <math>\operatorname{Int}_X f^{-1}(S) = f^{-1}\left(\operatorname{Int}_Y S\right)</math> and moreover, <math>S</math> is a regular open (resp. a regular closed)<ref group=note name="DefOfRegularOpenClosed" /> subset of <math>Y</math> if and only if <math>f^{-1}(S)</math> is a regular open (resp. a regular closed) subset of <math>X.</math> 
</li>
<li>If a [[Net (mathematics)|net]] <math>y_{\bull} = \left(y_i\right)_{i \in I}</math> [[Convergent net|converges]] in <math>Y</math> to a point <math>y \in Y</math> and if the continuous open map <math>f : X \to Y</math> is surjective, then for any <math>x \in f^{-1}(y)</math> there exists a net <math>x_{\bull} = \left(x_a\right)_{a \in A}</math> in <math>X</math> (indexed by some [[directed set]] <math>A</math>) such that <math>x_{\bull} \to x</math> in <math>X</math> and <math>f\left(x_{\bull}\right) := \left(f\left(x_a\right)\right)_{a \in A}</math> is a [[Subnet (mathematics)|subnet]] of <math>y_{\bull}.</math> Moreover, the indexing set <math>A</math> may be taken to be <math>A := I \times \mathcal{N}_x</math> with the [[product order]] where <math>\mathcal{N}_x</math> is any [[neighbourhood basis]] of <math>x</math> directed by <math>\,\supseteq.\,</math><ref group=note>Explicitly, for any <math>a := (i, U) \in A := I \times \mathcal{N}_x,</math> pick any <math>h_a \in I</math> such that <math>i \leq h_a \text{ and } y_{h_a} \in f(U)</math> and then let <math>x_a \in U \cap f^{-1}\left(y_{h_a}\right)</math> be arbitrary. The assignment <math>a \mapsto h_a</math> defines an [[order morphism]] <math>h : A \to I</math> such that <math>h(A)</math> is a [[cofinal subset]] of <math>I;</math> thus <math>f\left(x_{\bull}\right)</math> is a [[Willard-subnet]] of <math>y_{\bull}.</math></ref></li>
</ul>