Almost open map

Given a surjective map ${\displaystyle f:X\to Y,}$ a point ${\displaystyle x\in X}$ is called a point of openness for ${\displaystyle f}$ and ${\displaystyle f}$ is said to be open at ${\displaystyle x}$ (or an open map at ${\displaystyle x}$) if for every open neighborhood ${\displaystyle U}$ of ${\displaystyle x,}$ ${\displaystyle f(U)}$ is a neighborhood of ${\displaystyle f(x)}$ in ${\displaystyle Y}$ (note that the neighborhood ${\displaystyle f(U)}$ 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 each of its fibers has some point of openness. Explicitly, a surjective map ${\displaystyle f:X\to Y}$ is said to be almost open if for every ${\displaystyle y\in Y,}$ there exists some ${\displaystyle x\in f^{-1}(y)}$ such that ${\displaystyle f}$ is open at ${\displaystyle x.}$ Every almost open surjection is necessarily a pseudo-open map (introduced by Alexander Arhangelskii in 1963), which by definition means that for every ${\displaystyle y\in Y}$ and every neighborhood ${\displaystyle U}$ of ${\displaystyle f^{-1}(y)}$ (that is, ${\displaystyle f^{-1}(y)\subseteq \operatorname {Int} _{X}U}$), ${\displaystyle f(U)}$ is necessarily a neighborhood of ${\displaystyle y.}$

Every surjective open map is an almost open map but in general, the converse is not necessarily true. If a surjection ${\displaystyle f:(X,\tau )\to (Y,\sigma )}$ is an almost open map then it will be an open map if it satisfies the following condition (a condition that does not depend in any way on ${\displaystyle Y}$'s topology ${\displaystyle \sigma }$):

whenever ${\displaystyle m,n\in X}$ belong to the same fiber of ${\displaystyle f}$ (that is, ${\displaystyle f(m)=f(n)}$) then for every neighborhood ${\displaystyle U\in \tau }$ of ${\displaystyle m,}$ there exists some neighborhood ${\displaystyle V\in \tau }$ of ${\displaystyle n}$ such that ${\displaystyle F(V)\subseteq F(U).}$

If the map is continuous then the above condition is also necessary for the map to be open. That is, if ${\displaystyle f:X\to Y}$ is a continuous surjection then it is an open map if and only if it is almost open and it satisfies the above condition.

• Almost open set – Difference of an open set by a meager setPages displaying short descriptions of redirect targets
• Barrelled space – Type of topological vector space
• Bounded inverse theorem
• Closed graph – Graph of a map closed in the product spacePages displaying short descriptions of redirect targets
• Closed graph theorem – Theorem relating continuity to graphs
• Open set – Basic subset of a topological space
• Open and closed maps – A function that sends open (resp. closed) subsets to open (resp. closed) subsets
• Open mapping theorem (functional analysis) – Condition for a linear operator to be open (also known as the Banach–Schauder theorem)
• Quasi-open map – Function that maps non-empty open sets to sets that have non-empty interior in its codomain
• Surjection of Fréchet spaces – Characterization of surjectivity
• Webbed space – Space where open mapping and closed graph theorems hold

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