Английская Википедия:Almost open map
Шаблон:Short description Шаблон:More footnotes
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, Шаблон:Em 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 Шаблон:Em for <math>f</math> and <math>f</math> is said to be Шаблон:Em (or Шаблон:Em) if for every open neighborhood <math>U</math> of <math>x,</math> <math>f(U)</math> is a neighborhood of <math>f(x)</math> in <math>Y</math> (note that the neighborhood <math>f(U)</math> is not required to be an Шаблон:Em neighborhood).
A surjective map is called an Шаблон:Em if it is open at every point of its domain, while it is called an Шаблон:Em each of its fibers has some point of openness. Explicitly, a surjective map <math>f : X \to Y</math> is said to be Шаблон:Em 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 Шаблон:Em (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
A linear map <math>T : X \to Y</math> between two topological vector spaces (TVSs) is called a Шаблон:Em or an Шаблон:Em 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.Шаблон:Sfn
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.Шаблон: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 <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 Шаблон:Em 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 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:Шаблон:Sfn If <math>T : X \to Y</math> is a surjective linear operator from a locally convex space <math>X</math> onto a barrelled space <math>Y</math> then <math>T</math> is almost open.
- Theorem:Шаблон:Sfn 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 almost open.
The two theorems above do Шаблон:Em require the surjective linear map to satisfy Шаблон:Em topological conditions.
- Theorem:Шаблон:Sfn If <math>X</math> is a complete pseudometrizable TVS, <math>Y</math> is a Hausdorff TVS, and <math>T : X \to Y</math> is a closed and almost open linear surjection, then <math>T</math> is an open map.
- Theorem:Шаблон:Sfn Suppose <math>T : X \to Y</math> is a continuous linear operator from a complete pseudometrizable TVS <math>X</math> into a Hausdorff TVS <math>Y.</math> If the image of <math>T</math> is non-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
- Шаблон:Annotated link
- Шаблон:Annotated link
- Шаблон:Annotated link
- Шаблон:Annotated link
- Шаблон:Annotated link
- Шаблон:Annotated link
- Шаблон:Annotated link
- Шаблон:Annotated link (also known as the Banach–Schauder theorem)
- Шаблон:Annotated link
- Шаблон:Annotated link
- Шаблон:Annotated link
References
Bibliography
- Шаблон:Bourbaki Topological Vector Spaces Part 1 Chapters 1–5
- Шаблон:Husain Khaleelulla Barrelledness in Topological and Ordered Vector Spaces
- Шаблон:Jarchow Locally Convex Spaces
- Шаблон:Khaleelulla Counterexamples in Topological Vector Spaces
- Шаблон:Köthe Topological Vector Spaces I
- Шаблон:Narici Beckenstein Topological Vector Spaces
- Шаблон:Robertson Topological Vector Spaces
- Шаблон:Schaefer Wolff Topological Vector Spaces
- Шаблон:Trèves François Topological vector spaces, distributions and kernels
- Шаблон:Wilansky Modern Methods in Topological Vector Spaces
Шаблон:Functional Analysis Шаблон:BoundednessAndBornology Шаблон:TopologicalVectorSpaces
