View source for Almost open map

{{Short description|Map that satisfies a condition similar to that of being an open map.}}
{{more footnotes|date=June 2020}}

In [[functional analysis]] and related areas of [[mathematics]], an '''almost open map''' between [[topological spaces]] is a [[Function (mathematics)|map]] that satisfies a condition similar to, but weaker than, the condition of being an [[open map]]. 
As described below, for certain broad categories of [[topological vector spaces]], {{em|all}} [[Surjective map|surjective]] linear operators are necessarily almost open.

==Definitions==

Given a surjective map <math>f : X \to Y,</math> a point <math>x \in X</math> is called a {{em|{{visible anchor|point of openness}}}} for <math>f</math> and <math>f</math> is said to be {{em|open at <math>x</math>}} (or {{em|an open map at <math>x</math>}}) if 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).

A surjective map is called an {{em|[[open map]]}} if it is open at every point of its domain, while it is called an {{em|almost open map}} each of its [[Fiber (mathematics)|fibers]] has some point of openness. 
Explicitly, a surjective map <math>f : X \to Y</math> is said to be {{em|almost open}} if for every <math>y \in Y,</math> there exists some <math>x \in f^{-1}(y)</math> such that <math>f</math> is open at <math>x.</math> 
Every almost open surjection is necessarily a {{em|{{visible anchor|pseudo-open map}}}} (introduced by [[Alexander Arhangelskii]] in 1963), which by definition means that for every <math>y \in Y</math> and every neighborhood <math>U</math> of <math>f^{-1}(y)</math> (that is, <math>f^{-1}(y) \subseteq \operatorname{Int}_X U</math>), <math>f(U)</math> is necessarily a neighborhood of <math>y.</math>

===Almost open linear map===
{{anchor|Almost open linear map}}

A linear map <math>T : X \to Y</math> between two [[topological vector space]]s (TVSs) is called a {{em|{{visible anchor|nearly open linear map}}}} or an {{em|almost open linear map}} if for any neighborhood <math>U</math> of <math>0</math> in <math>X,</math> the closure of <math>T(U)</math> in <math>Y</math> is a neighborhood of the origin. 
Importantly, some authors use a different definition of "almost open map" in which they instead require that the linear map <math>T</math> satisfy: for any neighborhood <math>U</math> of <math>0</math> in <math>X,</math> the closure of <math>T(U)</math> in <math>T(X)</math> (rather than in <math>Y</math>) is a neighborhood of the origin; 
this article will not use this definition.{{sfn|Narici|Beckenstein|2011|pp=466-468}}

If a linear map <math>T : X \to Y</math> is almost open then because <math>T(X)</math> is a vector subspace of <math>Y</math> that contains a neighborhood of the origin in <math>Y,</math> the map <math>T : X \to Y</math> is necessarily [[surjective]]. 
For this reason many authors require surjectivity as part of the definition of "almost open". 

If <math>T : X \to Y</math> is a bijective linear operator, then <math>T</math> is almost open if and only if <math>T^{-1}</math> is [[almost continuous]].{{sfn|Narici|Beckenstein|2011|pp=466-468}}

==Relationship to open maps==

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.

===Open mapping theorems===

:'''Theorem''':{{sfn|Narici|Beckenstein|2011|pp=466-468}} If <math>T : X \to Y</math> is a surjective linear operator from a [[Locally convex topological vector space|locally convex]] space <math>X</math> onto a [[barrelled space]] <math>Y</math> then <math>T</math> is [[#nearly open linear map|almost open]].

:'''Theorem''':{{sfn|Narici|Beckenstein|2011|pp=466-468}} If <math>T : X \to Y</math> is a surjective linear operator from a TVS <math>X</math> onto a [[Baire space]] <math>Y</math> then <math>T</math> is [[#nearly open linear map|almost open]].

The two theorems above do {{em|not}} require the surjective linear map to satisfy {{em|any}} topological conditions.

:'''Theorem''':{{sfn|Narici|Beckenstein|2011|pp=466-468}} If <math>X</math> is a complete [[Metrizable topological vector space|pseudometrizable TVS]], <math>Y</math> is a Hausdorff TVS, and <math>T : X \to Y</math> is a closed and [[#nearly open linear map|almost open]] linear surjection, then <math>T</math> is an open map.

:'''Theorem''':{{sfn|Narici|Beckenstein|2011|pp=466-468}} Suppose <math>T : X \to Y</math> is a continuous linear operator from a complete [[Metrizable topological vector space|pseudometrizable TVS]] <math>X</math> into a Hausdorff TVS <math>Y.</math> If the image of <math>T</math> is non-[[Meager set|meager]] in <math>Y</math> then <math>T : X \to Y</math> is a surjective open map and <math>Y</math> is a complete metrizable space.

==See also==

* {{annotated link|Almost open set}}
* {{annotated link|Barrelled space}}
* {{annotated link|Bounded inverse theorem}}
* {{annotated link|Closed graph}}
* {{annotated link|Closed graph theorem}}
* {{annotated link|Open set}}
* {{annotated link|Open and closed maps}}
* {{annotated link|Open mapping theorem (functional analysis)}} (also known as the Banach–Schauder theorem)
* {{annotated link|Quasi-open map}}
* {{annotated link|Surjection of Fréchet spaces}}
* {{annotated link|Webbed space}}

==References==
{{reflist}}

==Bibliography==

* {{Bourbaki Topological Vector Spaces Part 1 Chapters 1–5}} <!-- {{sfn|Bourbaki|1987|p=}} -->
* {{Husain Khaleelulla Barrelledness in Topological and Ordered Vector Spaces}} <!-- {{sfn|Husain|Khaleelulla|1978|p=}} -->
* {{Jarchow Locally Convex Spaces}} <!-- {{sfn|Jarchow|1981|p=}} -->
* {{Khaleelulla Counterexamples in Topological Vector Spaces}} <!-- {{sfn|Khaleelulla|{{{year| 1982 }}}|p=}} --> 
* {{Köthe Topological Vector Spaces I}} <!-- {{sfn|Köthe|1969|p=}} -->
* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn|Narici|Beckenstein|2011|p=}} -->
* {{Robertson Topological Vector Spaces}} <!-- {{sfn|Robertson|Robertson|1980|p=}} -->
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!-- {{sfn|Schaefer|Wolff|1999|p=}} -->
* {{Trèves François Topological vector spaces, distributions and kernels}} <!-- {{sfn|Treves|2006|p=}} -->
* {{Wilansky Modern Methods in Topological Vector Spaces|edition=1}} <!-- {{sfn|Wilansky|2013|p=}} -->

{{Functional Analysis}}
{{BoundednessAndBornology}}
{{TopologicalVectorSpaces}}

[[Category:Topological vector spaces]]