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, all surjective linear operators are necessarily almost open.
Given a surjective map
is called a point of openness for
(or an open map at
) if for every open neighborhood
(note that the neighborhood
is not required to be an open neighborhood).
A surjective map is called an open map if it is open at every point of its domain, while it is called an almost open map if each of its fibers has some point of openness.
Explicitly, a surjective map
Every almost open surjection is necessarily a pseudo-open map (introduced by Alexander Arhangelskii in 1963), which by definition means that for every
is necessarily a neighborhood of
A linear map
between two topological vector spaces (TVSs) is called a nearly open linear map or an almost open linear map if for any neighborhood
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
satisfy: for any neighborhood
) is a neighborhood of the origin; this article will not use this definition.
[1] If a linear map
is a vector subspace of
that contains a neighborhood of the origin in
is necessarily surjective.
For this reason many authors require surjectivity as part of the definition of "almost open".
is a bijective linear operator, then
is almost continuous.
[1] Every surjective open map is an almost open map but in general, the converse is not necessarily true.
is an almost open map then it will be an open map if it satisfies the following condition (a condition that does not depend in any way on
): If the map is continuous then the above condition is also necessary for the map to be open.
is a continuous surjection then it is an open map if and only if it is almost open and it satisfies the above condition.
The two theorems above do not require the surjective linear map to satisfy any topological conditions.