Almost open map

In functional analysis and related areas of mathematics, an almost open map between topological spacess is a map that satisfies a condition similar to, but weaker than, the condition of being an open map.
Definitions.
Given a surjective map formula_1 a point formula_2 is called a ' for formula_3 and formula_3 is said to be ' (or ) if for every open neighborhood formula_5 of formula_6 formula_7 is a neighborhood of formula_8 in formula_9 (note that the neighborhood formula_7 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 formula_11 is said to be if for every formula_12 there exists some formula_13 such that formula_3 is open at formula_15 
Every almost open surjection is necessarily a (introduced by Alexander Arhangelskii in 1963), which by definition means that for every formula_16 and every neighborhood formula_5 of formula_18 (i.e. formula_19), formula_7 is necessarily a neighborhood of formula_21 
Almost open linear map.
A linear map formula_22 between two topological vector spaces (TVSs) is called almost open if for any neighborhood formula_5 of formula_24 in formula_25 the closure of formula_26 in formula_9 is a neighborhood of the origin.
Importantly, some authors call formula_28 is almost open if for any neighborhood formula_5 of formula_24 in formula_25 the closure of formula_26 in formula_33 (rather than in formula_9) is a neighborhood of the origin; 
this article will not use this definition.
If a linear operator formula_22 is almost open then because formula_33 is a vector subspace of formula_9 that contains a neighborhood of 0 in formula_22 is necessarily surjective. 
For this reason many authors require surjectivity as part of the definition of "almost open". 
If formula_22 is a bijective linear operator, then formula_28 is almost open if and only if formula_41 is almost continuous.
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 formula_42 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 formula_9's topology formula_44): 
If the map is continuous then the above condition is also necessary for the map to be open. That is, if formula_11 is a continuous surjection then it is an open map if and only if it is almost open and it satisfies the above condition.