Almost 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, Template:Em surjective linear operators are necessarily almost open.

Definitions

Given a surjective map $f:X\to Y,$ a point $x\in X$ is called a Template:Em for $f$ and $f$ is said to be Template:Em (or Template:Em) 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 Template:Em neighborhood).

A surjective map is called an Template:Em if it is open at every point of its domain, while it is called an Template:Em if each of its fibers has some point of openness. Explicitly, a surjective map $f:X\to Y$ is said to be Template:Em 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 Template:Em (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, $f^{-1}(y)\subseteq {\mathrm{Int}}_{X}U$), $f(U)$ 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 Template:Em or an Template:Em 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.Template:Sfn

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.Template:Sfn

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 Template:Em 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:Template:Sfn 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:Template:Sfn 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 Template:Em require the surjective linear map to satisfy Template:Em topological conditions.

Theorem:Template:Sfn 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:Template:Sfn 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.

See also

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Bibliography

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