Inmathematics, more specifically intopology, anopen map is afunction between twotopological spaces that mapsopen sets to open sets.^[1][2][3] That is, a function${\displaystyle f:X\to Y}$ is open if for any open set${\displaystyle U}$ in${\displaystyle X,}$ theimage${\displaystyle f(U)}$ is open in${\displaystyle Y.}$ Likewise, aclosed map is a function that mapsclosed sets to closed sets.^[3][4] A map may be open, closed, both, or neither;^[5] in particular, an open map need not be closed and vice versa.^[6]

Open^[7] and closed^[8] maps are not necessarilycontinuous.^[4] Further, continuity is independent of openness and closedness in the general case and a continuous function may have one, both, or neither property;^[3] this fact remains true even if one restricts oneself to metric spaces.^[9] Although their definitions seem more natural, open and closed maps are much less important than continuous maps. Recall that, by definition, a function${\displaystyle f:X\to Y}$ is continuous if thepreimage of every open set of${\displaystyle Y}$ is open in${\displaystyle X.}$^[2] (Equivalently, if the preimage of every closed set of${\displaystyle Y}$ is closed in${\displaystyle X}$).

Early study of open maps was pioneered bySimion Stoilow andGordon Thomas Whyburn.^[10]

If${\displaystyle S}$ is a subset of a topological space then let${\displaystyle \overline{S}}$ and${\displaystyle \mathrm{Cl}S}$ (resp.${\displaystyle \mathrm{Int}S}$) denote theclosure (resp.interior) of${\displaystyle S}$ in that space. Let${\displaystyle f:X\to Y}$ be a function betweentopological spaces. If${\displaystyle S}$ is any set then${\displaystyle f(S):=\left\{f(s):s\in S\cap \mathrm{domain}f\right\}}$ is called the image of${\displaystyle S}$ under${\displaystyle f.}$

There are two different competing, but closely related, definitions of "open map" that are widely used, where both of these definitions can be summarized as: "it is a map that sends open sets to open sets." The following terminology is sometimes used to distinguish between the two definitions.

A map${\displaystyle f:X\to Y}$ is called a

• "Strongly open map" if whenever${\displaystyle U}$ is anopen subset of the domain${\displaystyle X}$ then${\displaystyle f(U)}$ is an open subset of${\displaystyle f}$'scodomain${\displaystyle Y.}$
• "Relatively open map" if whenever${\displaystyle U}$ is an open subset of the domain${\displaystyle X}$ then${\displaystyle f(U)}$ is an open subset of${\displaystyle f}$'simage${\displaystyle \mathrm{Im}f:=f(X),}$ where as usual, this set is endowed with thesubspace topology induced on it by${\displaystyle f}$'s codomain${\displaystyle Y.}$^[11]

Every strongly open map is a relatively open map. However, these definitions are not equivalent in general.

Warning: Many authors define "open map" to mean "relatively open map" (for example, The Encyclopedia of Mathematics) while others define "open map" to mean "strongly open map". In general, these definitions arenot equivalent so it is thus advisable to always check what definition of "open map" an author is using.

Asurjective map is relatively open if and only if it is strongly open; so for this important special case the definitions are equivalent. More generally, a map${\displaystyle f:X\to Y}$ is relatively open if and only if thesurjection${\displaystyle f:X\to f(X)}$ is a strongly open map.

Because${\displaystyle X}$ is always an open subset of${\displaystyle X,}$ the image${\displaystyle f(X)=\mathrm{Im}f}$ of a strongly open map${\displaystyle f:X\to Y}$ must be an open subset of its codomain${\displaystyle Y.}$ In fact, a relatively open map is a strongly open map if and only if its image is an open subset of its codomain. In summary,

A map is strongly open if and only if it is relatively open and its image is an open subset of its codomain.

By using this characterization, it is often straightforward to apply results involving one of these two definitions of "open map" to a situation involving the other definition.

The discussion above will also apply to closed maps if each instance of the word "open" is replaced with the word "closed".

A map${\displaystyle f:X\to Y}$ is called anopen map or astrongly open map if it satisfies any of the following equivalent conditions:

1. Definition:${\displaystyle f:X\to Y}$ maps open subsets of its domain to open subsets of its codomain; that is, for any open subset${\displaystyle U}$ of${\displaystyle X}$,${\displaystyle f(U)}$ is an open subset of${\displaystyle Y.}$
2. ${\displaystyle f:X\to Y}$ is a relatively open map and its image${\displaystyle \mathrm{Im}f:=f(X)}$ is an open subset of its codomain${\displaystyle Y.}$
3. For every${\displaystyle x\in X}$ and everyneighborhood${\displaystyle N}$ of${\displaystyle x}$ (however small),${\displaystyle f(N)}$ is a neighborhood of${\displaystyle f(x)}$. We can replace the first or both instances of the word "neighborhood" with "open neighborhood" in this condition and the result will still be an equivalent condition:
   • For every${\displaystyle x\in X}$ and every open neighborhood${\displaystyle N}$ of${\displaystyle x}$,${\displaystyle f(N)}$ is a neighborhood of${\displaystyle f(x)}$.
   • For every${\displaystyle x\in X}$ and every open neighborhood${\displaystyle N}$ of${\displaystyle x}$,${\displaystyle f(N)}$ is an open neighborhood of${\displaystyle f(x)}$.
4. ${\displaystyle f\left({\mathrm{Int}}_{X}A\right)\subseteq {\mathrm{Int}}_Y(f(A))}$ for all subsets${\displaystyle A}$ of${\displaystyle X,}$ where${\displaystyle \mathrm{Int}}$ denotes thetopological interior of the set.
5. Whenever${\displaystyle C}$ is aclosed subset of${\displaystyle X}$ then the set${\displaystyle \left\{y\in Y:f^{-1}(y)\subseteq C\right\}}$ is a closed subset of${\displaystyle Y.}$
   • This is a consequence of theidentity${\displaystyle f(X\setminus R)=Y\setminus \left\{y\in Y:f^{-1}(y)\subseteq R\right\},}$ which holds for all subsets${\displaystyle R\subseteq X.}$

If${\displaystyle \mathcal{B}}$ is abasis for${\displaystyle X}$ then the following can be appended to this list:

6. ${\displaystyle f}$ maps basic open sets to open sets in its codomain (that is, for any basic open set${\displaystyle B\in \mathcal{B},}$${\displaystyle f(B)}$ is an open subset of${\displaystyle Y}$).

A map${\displaystyle f:X\to Y}$ is called arelatively closed map if whenever${\displaystyle C}$ is aclosed subset of the domain${\displaystyle X}$ then${\displaystyle f(C)}$ is a closed subset of${\displaystyle f}$'simage${\displaystyle \mathrm{Im}f:=f(X),}$ where as usual, this set is endowed with thesubspace topology induced on it by${\displaystyle f}$'scodomain${\displaystyle Y.}$

A map${\displaystyle f:X\to Y}$ is called aclosed map or astrongly closed map if it satisfies any of the following equivalent conditions:

1. Definition:${\displaystyle f:X\to Y}$ maps closed subsets of its domain to closed subsets of its codomain; that is, for any closed subset${\displaystyle C}$ of${\displaystyle X,}$${\displaystyle f(C)}$ is a closed subset of${\displaystyle Y.}$
2. ${\displaystyle f:X\to Y}$ is a relatively closed map and its image${\displaystyle \mathrm{Im}f:=f(X)}$ is a closed subset of its codomain${\displaystyle Y.}$
3. ${\displaystyle \overline{f(A)}\subseteq f\left(\overline{A}\right)}$ for every subset${\displaystyle A\subseteq X.}$
4. ${\displaystyle \overline{f(C)}\subseteq f(C)}$ for every closed subset${\displaystyle C\subseteq X.}$
5. ${\displaystyle \overline{f(C)}=f(C)}$ for every closed subset${\displaystyle C\subseteq X.}$
6. Whenever${\displaystyle U}$ is an open subset of${\displaystyle X}$ then the set${\displaystyle \left\{y\in Y:f^{-1}(y)\subseteq U\right\}}$ is an open subset of${\displaystyle Y.}$
7. If${\displaystyle x_{\bullet }}$ is anet in${\displaystyle X}$ and${\displaystyle y\in Y}$ is a point such that${\displaystyle f\left(x_{\bullet }\right)\to y}$ in${\displaystyle Y,}$ then${\displaystyle x_{\bullet }}$ converges in${\displaystyle X}$ to the set${\displaystyle f^{-1}(y).}$
   • The convergence${\displaystyle x_{\bullet }\to f^{-1}(y)}$ means that every open subset of${\displaystyle X}$ that contains${\displaystyle f^{-1}(y)}$ will contain${\displaystyle x_j}$ for all sufficiently large indices${\displaystyle j.}$

Asurjective map is strongly closed if and only if it is relatively closed. So for this important special case, the two definitions are equivalent. By definition, the map${\displaystyle f:X\to Y}$ is a relatively closed map if and only if thesurjection${\displaystyle f:X\to \mathrm{Im}f}$ is a strongly closed map.

If in the open set definition of "continuous map" (which is the statement: "every preimage of an open set is open"), both instances of the word "open" are replaced with "closed" then the statement of results ("every preimage of a closed set is closed") isequivalent to continuity. This does not happen with the definition of "open map" (which is: "every image of an open set is open") since the statement that results ("every image of a closed set is closed") is the definition of "closed map", which is in generalnot equivalent to openness. There exist open maps that are not closed and there also exist closed maps that are not open. This difference between open/closed maps and continuous maps is ultimately due to the fact that for any set${\displaystyle S,}$ only${\displaystyle f(X\setminus S)\supseteq f(X)\setminus f(S)}$ is guaranteed in general, whereas for preimages, equality${\displaystyle f^{-1}(Y\setminus S)=f^{-1}(Y)\setminus f^{-1}(S)}$ always holds.

The function${\displaystyle f:\mathbb{R}\to \mathbb{R}}$ defined by${\displaystyle f(x)=x^2}$ is continuous, closed, and relatively open, but not (strongly) open. This is because if${\displaystyle U=(a,b)}$ is any open interval in${\displaystyle f}$'s domain${\displaystyle \mathbb{R}}$ that doesnot contain${\displaystyle 0}$ then${\displaystyle f(U)=(min\{a^2,b^2\},max\{a^2,b^2\}),}$ where this open interval is an open subset of both${\displaystyle \mathbb{R}}$ and${\displaystyle \mathrm{Im}f:=f(\mathbb{R})=[0,\mathrm{\infty }).}$ However, if${\displaystyle U=(a,b)}$ is any open interval in${\displaystyle \mathbb{R}}$ that contains${\displaystyle 0}$ then${\displaystyle f(U)=[0,max\{a^2,b^2\}),}$ which is not an open subset of${\displaystyle f}$'s codomain${\displaystyle \mathbb{R}}$ butis an open subset of${\displaystyle \mathrm{Im}f=[0,\mathrm{\infty }).}$ Because the set of all open intervals in${\displaystyle \mathbb{R}}$ is abasis for theEuclidean topology on${\displaystyle \mathbb{R},}$ this shows that${\displaystyle f:\mathbb{R}\to \mathbb{R}}$ is relatively open but not (strongly) open.

If${\displaystyle Y}$ has thediscrete topology (that is, all subsets are open and closed) then every function${\displaystyle f:X\to Y}$ is both open and closed (but not necessarily continuous). For example, thefloor function from${\displaystyle \mathbb{R}}$ to${\displaystyle \mathbb{Z}}$ is open and closed, but not continuous. This example shows that the image of aconnected space under an open or closed map need not be connected.

Whenever we have aproduct of topological spaces$X=\prod X_i,$ the natural projections${\displaystyle p_i:X\to X_i}$ are open^[12][13] (as well as continuous). Since the projections offiber bundles andcovering maps are locally natural projections of products, these are also open maps. Projections need not be closed however. Consider for instance the projection${\displaystyle p_1:{\mathbb{R}}^2\to \mathbb{R}}$ on the first component; then the set${\displaystyle A=\{(x,1/x):x\ne 0\}}$ is closed in${\displaystyle {\mathbb{R}}^2,}$ but${\displaystyle p_1(A)=\mathbb{R}\setminus \{0\}}$ is not closed in${\displaystyle \mathbb{R}.}$ However, for a compact space${\displaystyle Y,}$ the projection${\displaystyle X\times Y\to X}$ is closed. This is essentially thetube lemma.

To every point on theunit circle we can associate theangle of the positive${\displaystyle x}$-axis with the ray connecting the point with the origin. This function from the unit circle to the half-openinterval [0,2π) is bijective, open, and closed, but not continuous. It shows that the image of acompact space under an open or closed map need not be compact. Also note that if we consider this as a function from the unit circle to the real numbers, then it is neither open nor closed. Specifying thecodomain is essential.

Everyhomeomorphism is open, closed, and continuous. In fact, abijective continuous map is a homeomorphismif and only if it is open, or equivalently, if and only if it is closed.

Thecomposition of two (strongly) open maps is an open map and the composition of two (strongly) closed maps is a closed map.^[14][15] However, the composition of two relatively open maps need not be relatively open and similarly, the composition of two relatively closed maps need not be relatively closed. If${\displaystyle f:X\to Y}$ is strongly open (respectively, strongly closed) and${\displaystyle g:Y\to Z}$ is relatively open (respectively, relatively closed) then${\displaystyle g\circ f:X\to Z}$ is relatively open (respectively, relatively closed).

Let${\displaystyle f:X\to Y}$ be a map. Given any subset${\displaystyle T\subseteq Y,}$ if${\displaystyle f:X\to Y}$ is a relatively open (respectively, relatively closed, strongly open, strongly closed, continuous,surjective) map then the same is true of its restriction$${\displaystyle f{|}_{f^{-1}(T)}:f^{-1}(T)\to T}$$to the${\displaystyle f}$-saturated subset${\displaystyle f^{-1}(T).}$

The categorical sum of two open maps is open, or of two closed maps is closed.^[15] The categoricalproduct of two open maps is open, however, the categorical product of two closed maps need not be closed.^[14][15]

A bijective map is open if and only if it is closed. The inverse of a bijective continuous map is a bijective open/closed map (and vice versa). A surjective open map is not necessarily a closed map, and likewise, a surjective closed map is not necessarily an open map. Alllocal homeomorphisms, including allcoordinate charts onmanifolds and allcovering maps, are open maps.

A variant of the closed map lemma states that if a continuous function betweenlocally compact Hausdorff spaces is proper then it is also closed.

Incomplex analysis, the identically namedopen mapping theorem states that every non-constantholomorphic function defined on aconnected open subset of thecomplex plane is an open map.

Theinvariance of domain theorem states that a continuous and locally injective function between two${\displaystyle n}$-dimensionaltopological manifolds must be open.

Infunctional analysis, theopen mapping theorem states that every surjective continuouslinear operator betweenBanach spaces is an open map. This theorem has been generalized totopological vector spaces beyond just Banach spaces.

A surjective map${\displaystyle f:X\to Y}$ is called analmost open map if for every${\displaystyle y\in Y}$ there exists some${\displaystyle x\in f^{-1}(y)}$ such that${\displaystyle x}$ is apoint of openness for${\displaystyle f,}$ which by definition means that 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). 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 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.

If${\displaystyle f:X\to Y}$ is a continuous map that is also openor closed then:

• if${\displaystyle f}$ is a surjection then it is aquotient map and even ahereditarily quotient map,
  • A surjective map${\displaystyle f:X\to Y}$ is calledhereditarily quotient if for every subset${\displaystyle T\subseteq Y,}$ the restriction${\displaystyle f{|}_{f^{-1}(T)}:f^{-1}(T)\to T}$ is a quotient map.
• if${\displaystyle f}$ is aninjection then it is atopological embedding.
• if${\displaystyle f}$ is abijection then it is ahomeomorphism.

In the first two cases, being open or closed is merely asufficient condition for the conclusion that follows. In the third case, it isnecessary as well.

If${\displaystyle f:X\to Y}$ is a continuous (strongly) open map,${\displaystyle A\subseteq X,}$ and${\displaystyle S\subseteq Y,}$ then:

• ${\displaystyle f^{-1}\left({\mathrm{Bd}}_{Y}S\right)={\mathrm{Bd}}_X\left(f^{-1}(S)\right)}$ where${\displaystyle \mathrm{Bd}}$ denotes theboundary of a set.
• ${\displaystyle f^{-1}\left(\overline{S}\right)=\overline{f^{-1}(S)}}$ where${\displaystyle \overline{S}}$ denote theclosure of a set.
• If${\displaystyle \overline{A}=\overline{{\mathrm{Int}}_{X}A},}$ where${\displaystyle \mathrm{Int}}$ denotes theinterior of a set, then$${\displaystyle \overline{{\mathrm{Int}}_{Y}f(A)}=\overline{f(A)}=\overline{f\left({\mathrm{Int}}_{X}A\right)}=\overline{f\left(\overline{{\mathrm{Int}}_{X}A}\right)}}$$where this set${\displaystyle \overline{f(A)}}$ is also necessarily aregular closed set (in${\displaystyle Y}$).^{[note 1]} In particular, if${\displaystyle A}$ is a regular closed set then so is${\displaystyle \overline{f(A)}.}$ And if${\displaystyle A}$ is aregular open set then so is${\displaystyle Y\setminus \overline{f(X\setminus A)}.}$
• If the continuous open map${\displaystyle f:X\to Y}$ is also surjective then${\displaystyle {\mathrm{Int}}_X{f}^{-1}(S)=f^{-1}\left({\mathrm{Int}}_{Y}S\right)}$ and moreover,${\displaystyle S}$ is a regular open (resp. a regular closed)^{[note 1]} subset of${\displaystyle Y}$ if and only if${\displaystyle f^{-1}(S)}$ is a regular open (resp. a regular closed) subset of${\displaystyle X.}$
• If anet${\displaystyle y_{\bullet }={\left(y_i\right)}_{i\in I}}$converges in${\displaystyle Y}$ to a point${\displaystyle y\in Y}$ and if the continuous open map${\displaystyle f:X\to Y}$ is surjective, then for any${\displaystyle x\in f^{-1}(y)}$ there exists a net${\displaystyle x_{\bullet }={\left(x_a\right)}_{a\in A}}$ in${\displaystyle X}$ (indexed by somedirected set${\displaystyle A}$) such that${\displaystyle x_{\bullet }\to x}$ in${\displaystyle X}$ and${\displaystyle f\left(x_{\bullet }\right):={\left(f\left(x_a\right)\right)}_{a\in A}}$ is asubnet of${\displaystyle y_{\bullet }.}$ Moreover, the indexing set${\displaystyle A}$ may be taken to be${\displaystyle A:=I\times {\mathcal{N}}_x}$ with theproduct order where${\displaystyle {\mathcal{N}}_x}$ is anyneighbourhood basis of${\displaystyle x}$ directed by${\displaystyle \supseteq .}$^{[note 2]}

• Almost open map – Map that satisfies a condition similar to that of being an open map.
• Closed graph – Graph of a map closed in the product spacePages displaying short descriptions of redirect targets
• Closed linear operator – Linear operator whose graph is closed
• Local homeomorphism – Mathematical function revertible near each point
• Quasi-open map – Function that maps non-empty open sets to sets that have non-empty interior in its codomain
• Quotient map (topology) – Topological space constructionPages displaying short descriptions of redirect targets
• Perfect map – Continuous closed surjective map, each of whose fibers are also compact sets
• Proper map – Map between topological spaces with the property that the preimage of every compact is compact
• Sequence covering map

• Narici, Lawrence; Beckenstein, Edward (2011).Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press.ISBN 978-1584888666.OCLC 144216834.
• 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.

• General topology
• Theory of continuous functions
• Lemmas

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