Editing Open and closed maps (section)

==Sufficient conditions==

Every [[homeomorphism]] is open, closed, and continuous. In fact, a [[bijective]] continuous map is a homeomorphism [[if and only if]] it is open, or equivalently, if and only if it is closed. 

The [[Function composition|composition]] of two (strongly) open maps is an open map and the composition of two (strongly) closed maps is a closed map.<ref name=baues55>{{cite book|last1=Baues|first1=Hans-Joachim|last2=Quintero|first2=Antonio|date=2001|title=Infinite Homotopy Theory|series=''K''-Monographs in Mathematics|volume=6|isbn=9780792369820|page=53|quote=A composite of open maps is open and a composite of closed maps is closed. Also, a product of open maps is open. In contrast, a product of closed maps is not necessarily closed,...}}</ref><ref name=james49>{{cite book|last=James|first=I. M.|date=1984|title=General Topology and Homotopy Theory|url=https://archive.org/details/generaltopologyh00imja|url-access=limited|publisher=Springer-Verlag|isbn=9781461382836|page=[https://archive.org/details/generaltopologyh00imja/page/n56 49]|quote=...let us recall that the composition of open maps is open and the composition of closed maps is closed. Also that the sum of open maps is open and the sum of closed maps is closed. However, the product of closed maps is not necessarily closed, although the product of open maps is open.}}</ref> However, the composition of two relatively open maps need not be relatively open and similarly, the composition of two relatively closed maps need not be relatively closed.  
If <math>f : X \to Y</math> is strongly open (respectively, strongly closed) and <math>g : Y \to Z</math> is relatively open (respectively, relatively closed) then <math>g \circ f : X \to Z</math> is relatively open (respectively, relatively closed).  

Let <math>f : X \to Y</math> be a map. 
Given any subset <math>T \subseteq Y,</math> if <math>f : X \to Y</math> is a relatively open (respectively, relatively closed, strongly open, strongly closed, continuous, [[Surjective function|surjective]]) map then the same is true of its restriction 
<math display=block>f\big\vert_{f^{-1}(T)} ~:~ f^{-1}(T) \to T</math>
to the [[Saturated set|<math>f</math>-saturated]] subset <math>f^{-1}(T).</math>

The categorical sum of two open maps is open, or of two closed maps is closed.<ref name=james49/> 
The categorical [[Product (topology)|product]] of two open maps is open, however, the categorical product of two closed maps need not be closed.<ref name=baues55/><ref name=james49/> 

A bijective map is open if and only if it is closed. 
The inverse of a bijective continuous map is a bijective open/closed map (and vice versa). 
A surjective open map is not necessarily a closed map, and likewise, a surjective closed map is not necessarily an open map. All [[local homeomorphism]]s, including all [[coordinate chart]]s on [[manifold]]s and all [[covering map]]s, are open maps. 

{{Math theorem|name=Closed map lemma|math_statement=
Every continuous function <math>f : X \to Y</math> from a [[compact space]] <math>X</math> to a [[Hausdorff space]] <math>Y</math> is closed and [[Proper map|proper]] (meaning that preimages of compact sets are compact).
}}

A variant of the closed map lemma states that if a continuous function between [[Locally compact space|locally compact]] Hausdorff spaces is proper then it is also closed. 

In [[complex analysis]], the identically named [[Open mapping theorem (complex analysis)|open mapping theorem]] states that every non-constant [[holomorphic function]] defined on a [[Connected space|connected]] open subset of the [[complex plane]] is an open map. 

The [[invariance of domain]] theorem states that a continuous and locally injective function between two <math>n</math>-dimensional [[Manifold|topological manifolds]] must be open. 

{{Math theorem|name=[[Invariance of domain]]|math_statement=
If <math>U</math> is an [[Open set|open subset]] of <math>\R^n</math> and <math>f : U \to \R^n</math> is an [[injective]] [[continuous map]], then <math>V := f(U)</math> is open in <math>\R^n</math> and <math>f</math> is a [[homeomorphism]] between <math>U</math> and <math>V.</math>
}}

In [[functional analysis]], the [[Open mapping theorem (functional analysis)|open mapping theorem]] states that every surjective continuous [[linear operator]] between [[Banach space]]s is an open map. 
This theorem has been generalized to [[topological vector space]]s beyond just Banach spaces. 

A surjective map <math>f : X \to Y</math> is called an '''{{em|[[almost open map]]}}'''{{anchor|Almost open map}} if for every <math>y \in Y</math> there exists some <math>x \in f^{-1}(y)</math> such that <math>x</math> is a '''{{em|{{visible anchor|point of openness|Point of openness}}}}''' for <math>f,</math> which by definition means that for every open neighborhood <math>U</math> of <math>x,</math> <math>f(U)</math> is a [[Neighborhood  (topology)|neighborhood]] of <math>f(x)</math> in <math>Y</math> (note that the neighborhood <math>f(U)</math> is not required to be an {{em|open}} neighborhood). 
Every surjective open map is an almost open map but in general, the converse is not necessarily true. 
If a surjection <math>f : (X, \tau) \to (Y, \sigma)</math> is an almost open map then it will be an open map if it satisfies the following condition (a condition that does {{em|not}} depend in any way on <math>Y</math>'s topology <math>\sigma</math>): 
:whenever <math>m, n \in X</math> belong to the same [[Fiber (mathematics)|fiber]] of <math>f</math> (that is, <math>f(m) = f(n)</math>) then for every neighborhood <math>U \in \tau</math> of <math>m,</math> there exists some neighborhood <math>V \in \tau</math> of <math>n</math> such that <math>F(V) \subseteq F(U).</math> 
If the map is continuous then the above condition is also necessary for the map to be open. That is, if <math>f : X \to Y</math> is a continuous surjection then it is an open map if and only if it is almost open and it satisfies the above condition.