almost open map

{{Short description|Map that satisfies a condition similar to that of being an open map.}}

In functional analysis and related areas of mathematics, an almost open map between topological spaces is a 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 linear operators are necessarily almost open.

Definitions

Given a surjective map $f : X \to Y,$ a point $x \in X$ is called a {{em|{{visible anchor|point of openness}}}} for $f$ and $f$ is said to be {{em|open at $x$ }} (or {{em|an open map at $x$ }}) if for every open neighborhood $U$ of $x,$ $f(U)$ is a neighborhood of $f(x)$ in $Y$ (note that the neighborhood $f(U)$ 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}} if each of its fibers has some point of openness.

Explicitly, a surjective map $f : X \to Y$ is said to be {{em|almost open}} if for every $y \in Y,$ there exists some $x \in f^{-1}(y)$ such that $f$ is open at $x.$

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 $y \in Y$ and every neighborhood $U$ of $f^{-1}(y)$ (that is, $f^{-1}(y) \subseteq \operatorname{Int}_X U$ ), $f(U)$ is necessarily a neighborhood of $y.$

=Almost open linear map=

A linear map $T : X \to Y$ between two topological vector spaces (TVSs) is called a {{em|{{visible anchor|nearly open linear map}}}} or an {{em|almost open linear map}} if for any neighborhood $U$ of $0$ in $X,$ the closure of $T(U)$ in $Y$ 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 $T$ satisfy: for any neighborhood $U$ of $0$ in $X,$ the closure of $T(U)$ in $T(X)$ (rather than in $Y$ ) is a neighborhood of the origin;

this article will not use this definition.{{sfn|Narici|Beckenstein|2011|pp=466-468}}

If a linear map $T : X \to Y$ is almost open then because $T(X)$ is a vector subspace of $Y$ that contains a neighborhood of the origin in $Y,$ the map $T : X \to Y$ is necessarily surjective.

For this reason many authors require surjectivity as part of the definition of "almost open".

If $T : X \to Y$ is a bijective linear operator, then $T$ is almost open if and only if $T^{-1}$ 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 $f : (X, \tau) \to (Y, \sigma)$ 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 $Y$ 's topology $\sigma$ ):

:whenever $m, n \in X$ belong to the same fiber of $f$ (that is, $f(m) = f(n)$ ) then for every neighborhood $U \in \tau$ of $m,$ there exists some neighborhood $V \in \tau$ of $n$ such that $F(V) \subseteq F(U).$

If the map is continuous then the above condition is also necessary for the map to be open. That is, if $f : X \to Y$ 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 $T : X \to Y$ is a surjective linear operator from a locally convex space $X$ onto a barrelled space $Y$ then $T$ is almost open.

:Theorem:{{sfn|Narici|Beckenstein|2011|pp=466-468}} If $T : X \to Y$ is a surjective linear operator from a TVS $X$ onto a Baire space $Y$ then $T$ is 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 $X$ is a complete pseudometrizable TVS, $Y$ is a Hausdorff TVS, and $T : X \to Y$ is a closed and almost open linear surjection, then $T$ is an open map.

:Theorem:{{sfn|Narici|Beckenstein|2011|pp=466-468}} Suppose $T : X \to Y$ is a continuous linear operator from a complete pseudometrizable TVS $X$ into a Hausdorff TVS $Y.$ If the image of $T$ is non-meager in $Y$ then $T : X \to Y$ is a surjective open map and $Y$ is a complete metrizable space.

References

Bibliography

{{Bourbaki Topological Vector Spaces Part 1 Chapters 1–5}}
{{Husain Khaleelulla Barrelledness in Topological and Ordered Vector Spaces}}
{{Jarchow Locally Convex Spaces}}
{{Khaleelulla Counterexamples in Topological Vector Spaces}}
{{Köthe Topological Vector Spaces I}}
{{Narici Beckenstein Topological Vector Spaces|edition=2}}
{{Robertson Topological Vector Spaces}}
{{Schaefer Wolff Topological Vector Spaces|edition=2}}
{{Trèves François Topological vector spaces, distributions and kernels}}
{{Wilansky Modern Methods in Topological Vector Spaces|edition=1}}

Category:Topological vector spaces