This article includes a list ofgeneral references, butit lacks sufficient correspondinginline citations. Please help toimprove this article byintroducing more precise citations.(June 2020) (Learn how and when to remove this message)

Infunctional analysis and related areas ofmathematics, analmost open map betweentopological spaces is amap that satisfies a condition similar to, but weaker than, the condition of being anopen map. As described below, for certain broad categories oftopological vector spaces,allsurjective linear operators are necessarily almost open.

Given a surjective map${\displaystyle f:X\to Y,}$ a point${\displaystyle x\in X}$ is called apoint of openness for${\displaystyle f}$ and${\displaystyle f}$ is said to beopen at${\displaystyle x}$ (oran open map at${\displaystyle x}$) if for every open neighborhood${\displaystyle U}$ of${\displaystyle x,}$${\displaystyle f(U)}$ is aneighborhood of${\displaystyle f(x)}$ in${\displaystyle Y}$ (note that the neighborhood${\displaystyle f(U)}$ is not required to be anopen neighborhood).

A surjective map is called anopen map if it is open at every point of its domain, while it is called analmost open map if each of itsfibers has some point of openness. Explicitly, a surjective map${\displaystyle f:X\to Y}$ is said to bealmost 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 apseudo-open map (introduced byAlexander 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,${\displaystyle f^{-1}(y)\subseteq {\mathrm{Int}}_{X}U}$),${\displaystyle f(U)}$ is necessarily a neighborhood of${\displaystyle y.}$

A linear map${\displaystyle T:X\to Y}$ between twotopological vector spaces (TVSs) is called anearly open linear map or analmost open linear map if for any neighborhood${\displaystyle U}$ of${\displaystyle 0}$ in${\displaystyle X,}$ the closure of${\displaystyle T(U)}$ in${\displaystyle Y}$ 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${\displaystyle T}$ satisfy: for any neighborhood${\displaystyle U}$ of${\displaystyle 0}$ in${\displaystyle X,}$ the closure of${\displaystyle T(U)}$ in${\displaystyle T(X)}$ (rather than in${\displaystyle Y}$) is a neighborhood of the origin; this article will not use this definition.^[1]

If a linear map${\displaystyle T:X\to Y}$ is almost open then because${\displaystyle T(X)}$ is a vector subspace of${\displaystyle Y}$ that contains a neighborhood of the origin in${\displaystyle Y,}$ the map${\displaystyle T:X\to Y}$ is necessarilysurjective. For this reason many authors require surjectivity as part of the definition of "almost open".

If${\displaystyle T:X\to Y}$ is a bijective linear operator, then${\displaystyle T}$ is almost open if and only if${\displaystyle T^{-1}}$ isalmost continuous.^[1]

Every surjectiveopen 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 doesnot depend in any way on${\displaystyle Y}$'s topology${\displaystyle \sigma }$):

whenever${\displaystyle m,n\in X}$ belong to the samefiber 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.

Theorem:^[1] If${\displaystyle T:X\to Y}$ is a surjective linear operator from alocally convex space${\displaystyle X}$ onto abarrelled space${\displaystyle Y}$ then${\displaystyle T}$ isalmost open.

Theorem:^[1] If${\displaystyle T:X\to Y}$ is a surjective linear operator from a TVS${\displaystyle X}$ onto aBaire space${\displaystyle Y}$ then${\displaystyle T}$ isalmost open.

The two theorems above donot require the surjective linear map to satisfyany topological conditions.

Theorem:^[1] If${\displaystyle X}$ is a completepseudometrizable TVS,${\displaystyle Y}$ is a Hausdorff TVS, and${\displaystyle T:X\to Y}$ is a closed andalmost open linear surjection, then${\displaystyle T}$ is an open map.

Theorem:^[1] Suppose${\displaystyle T:X\to Y}$ is a continuous linear operator from a completepseudometrizable TVS${\displaystyle X}$ into a Hausdorff TVS${\displaystyle Y.}$ If the image of${\displaystyle T}$ is non-meager in${\displaystyle Y}$ then${\displaystyle T:X\to Y}$ is a surjective open map and${\displaystyle Y}$ is a complete metrizable space.

• 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 – Condition for a linear operator to be openPages displaying short descriptions of redirect targets
• Closed graph – Property of functions in topologyPages 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

• Bourbaki, Nicolas (1987) [1981].Topological Vector Spaces: Chapters 1–5.Éléments de mathématique. Translated by Eggleston, H.G.; Madan, S. Berlin New York: Springer-Verlag.ISBN 3-540-13627-4.OCLC 17499190.
• Husain, Taqdir; Khaleelulla, S. M. (1978).Barrelledness in Topological and Ordered Vector Spaces.Lecture Notes in Mathematics. Vol. 692. Berlin, New York, Heidelberg:Springer-Verlag.ISBN 978-3-540-09096-0.OCLC 4493665.
• Jarchow, Hans (1981).Locally convex spaces. Stuttgart: B.G. Teubner.ISBN 978-3-519-02224-4.OCLC 8210342.
• Khaleelulla, S. M. (1982).Counterexamples in Topological Vector Spaces.Lecture Notes in Mathematics. Vol. 936. Berlin, Heidelberg, New York:Springer-Verlag.ISBN 978-3-540-11565-6.OCLC 8588370.
• Köthe, Gottfried (1983) [1969].Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. Vol. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media.ISBN 978-3-642-64988-2.MR 0248498.OCLC 840293704.
• Narici, Lawrence; Beckenstein, Edward (2011).Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press.ISBN 978-1584888666.OCLC 144216834.
• Robertson, Alex P.; Robertson, Wendy J. (1980).Topological Vector Spaces.Cambridge Tracts in Mathematics. Vol. 53. Cambridge England:Cambridge University Press.ISBN 978-0-521-29882-7.OCLC 589250.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999).Topological Vector Spaces.GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer.ISBN 978-1-4612-7155-0.OCLC 840278135.
• Trèves, François (2006) [1967].Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications.ISBN 978-0-486-45352-1.OCLC 853623322.
• Wilansky, Albert (2013).Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc.ISBN 978-0-486-49353-4.OCLC 849801114.

• Topological vector spaces

• Articles with short description
• Short description with empty Wikidata description
• Articles lacking in-text citations from June 2020
• All articles lacking in-text citations
• Pages displaying short descriptions of redirect targets via Module:Annotated link