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, surjective linear operators are necessarily almost open. Given a surjective map a point is called a for and is said to be (or ) if for every open neighborhood of is a neighborhood of in (note that the neighborhood is not required to be an neighborhood). A surjective map is called an if it is open at every point of its domain, while it is called an each of its fibers has some point of openness. Explicitly, a surjective map is said to be if for every there exists some such that is open at Every almost open surjection is necessarily a (introduced by Alexander Arhangelskii in 1963), which by definition means that for every and every neighborhood of (that is, ), is necessarily a neighborhood of A linear map between two topological vector spaces (TVSs) is called a or an if for any neighborhood of in the closure of in 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 of in the closure of in (rather than in ) is a neighborhood of the origin; this article will not use this definition. If a linear map is almost open then because is a vector subspace of that contains a neighborhood of the origin in the map is necessarily surjective. For this reason many authors require surjectivity as part of the definition of "almost open". If is a bijective linear operator, then is almost open if and only if is almost continuous. Every surjective open map is an almost open map but in general, the converse is not necessarily true. If a surjection is an almost open map then it will be an open map if it satisfies the following condition (a condition that does depend in any way on 's topology ): whenever belong to the same fiber of (that is, ) then for every neighborhood of there exists some neighborhood of such that If the map is continuous then the above condition is also necessary for the map to be open.