{"history":[],"article":{"id":"62feaf0f-64ea-4bdf-b8f2-968304e61ee5","title":"Open and closed maps","slug":"open-and-closed-maps","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), more specifically in [topology](https://bliptext.com/articles/topology), an **open map** is a [function](https://bliptext.com/articles/function-mathematics) between two [topological spaces](https://bliptext.com/articles/topological-space) that maps [open sets](https://bliptext.com/articles/open-set) to [open sets](https://bliptext.com/articles/open-set).\nThat is, a function f : X \\to Y is open if for any open set U in X, the [image](https://bliptext.com/articles/image-mathematics) f(U) is open in Y. Likewise, a **closed map** is a function that maps [closed sets](https://bliptext.com/articles/closed-set) to [closed sets](https://bliptext.com/articles/closed-set).\nA map may be open, closed, both, or neither; in particular, an open map need not be closed and vice versa.\nOpen and closed maps are not necessarily [continuous](https://bliptext.com/articles/continuous-function-topology).\nFurther, continuity is independent of openness and closedness in the general case and a continuous function may have one, both, or neither property; this fact remains true even if one restricts oneself to metric spaces.\nAlthough their definitions seem more natural, open and closed maps are much less important than continuous maps.\nRecall that, by definition, a function f : X \\to Y is continuous if the [preimage](https://bliptext.com/articles/preimage) of every open set of Y is open in X. (Equivalently, if the [preimage](https://bliptext.com/articles/preimage) of every closed set of Y is closed in X).\nEarly study of open maps was pioneered by [Simion Stoilow](https://bliptext.com/articles/simion-stoilow) and [Gordon Thomas Whyburn](https://bliptext.com/articles/gordon-thomas-whyburn).\n\n# Definitions and characterizations\nIf S is a subset of a topological space then let and (resp. ) denote the [closure](https://bliptext.com/articles/closure-topology) (resp. [interior](https://bliptext.com/articles/interior-topology)) of S in that space.\nLet f : X \\to Y be a function between [topological spaces](https://bliptext.com/articles/topological-space).\nIf S is any set then is called the image of S under f.\n\n## Competing definitions\nThere are two different competing, but closely related, definitions of \"\" that are widely used, where both of these definitions can be summarized as: \"it is a map that sends open sets to open sets.\"\nThe following terminology is sometimes used to distinguish between the two definitions.\nA map f : X \\to Y is called a\nEvery strongly open map is a relatively open map.\nHowever, these definitions are not equivalent in general.\nA [surjective](https://bliptext.com/articles/surjective-function) map is relatively open if and only if it is strongly open; so for this important special case the definitions are equivalent.\nMore generally, a map f : X \\to Y is relatively open if and only if the [surjection](https://bliptext.com/articles/surjective-function) is a strongly open map.\nBecause X is always an open subset of X, the image of a strongly open map f : X \\to Y must be an open subset of its codomain 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.\nIn summary,\nBy 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.\nThe discussion above will also apply to closed maps if each instance of the word \"open\" is replaced with the word \"closed\".\n\n## Open maps\nA map f : X \\to Y is called an **' or a **' if it satisfies any of the following equivalent conditions:\n

Definition: f : X \\to Y maps open subsets of its domain to open subsets of its codomain; that is, for any open subset U of X, f(U) is an open subset of Y.
f : X \\to Y is a relatively open map and its image is an open subset of its codomain Y.
For every x \\in X and every [neighborhood](https://bliptext.com/articles/neighborhood-topology) N of x (however small), f(N) is a [neighborhood](https://bliptext.com/articles/neighborhood-topology) of f(x).\nWe 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:\n
for all subsets A of X, where denotes the [topological interior](https://bliptext.com/articles/topological-interior) of the set.
Whenever C is a [closed subset](https://bliptext.com/articles/closed-set) of X then the set is a [closed subset](https://bliptext.com/articles/closed-set) of Y.

\nIf \\mathcal{B} is a [basis](https://bliptext.com/articles/base-topology) for X then the following can be appended to this list:\n\n## Closed maps\n****A**** ****m****a****p**** ****f**** ****:**** ****X**** ****\\****t****o**** ****Y**** ****i****s**** ****c****a****l****l****e****d**** ****a**** **** ****i****f**** ****w****h****e****n****e****v****e****r**** ****C**** ****i****s**** ****a**** ****c****l****o****s****e****d**** ****s****u****b****s****e****t**** ****o****f**** ****t****h****e**** ****d****o****m****a****i****n**** ****X**** ****t****h****e****n**** ****f****(****C****)**** ****i****s**** ****a**** ****c****l****o****s****e****d**** ****s****u****b****s****e****t**** ****o****f**** ****f****'****s**** ****i****m****a****g****e**** ****w****h****e****r****e**** ****a****s**** ****u****s****u****a****l****,**** ****t****h****i****s**** ****s****e****t**** ****i****s**** ****e****n****d****o****w****e****d**** ****w****i****t****h**** ****t****h****e**** ****s****u****b****s****p****a****c****e**** ****t****o****p****o****l****o****g****y**** ****i****n****d****u****c****e****d**** ****o****n**** ****i****t**** ****b****y**** ****f****'****s**** ****c****o****d****o****m****a****i****n**** ****Y****.****\nA map f : X \\to Y is called a **' or a **' if it satisfies any of the following equivalent conditions:\n

Definition: f : X \\to Y maps closed subsets of its domain to closed subsets of its codomain; that is, for any closed subset C of X, f(C) is a closed subset of Y.
f : X \\to Y is a relatively closed map and its image is a closed subset of its codomain Y.
for every subset
for every closed subset
for every closed subset
Whenever U is an open subset of X then the set is an open subset of Y.
If x_{\\bull} is a [net](https://bliptext.com/articles/net-mathematics) in X and y \\in Y is a point such that in Y, then x_{\\bull} converges in X to the set f^{-1}(y).\n

\nA [surjective](https://bliptext.com/articles/surjective-function) map is strongly closed if and only if it is relatively closed.\nSo for this important special case, the two definitions are equivalent.\nBy definition, the map f : X \\to Y is a relatively closed map if and only if the [surjection](https://bliptext.com/articles/surjective-function) is a strongly closed map.\nIf in the open set definition of \"[continuous map](https://bliptext.com/articles/continuous-function)\" (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\") is to continuity.\nThis 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 general equivalent to openness.\nThere exist open maps that are not closed and there also exist closed maps that are not open.\nThis difference between open/closed maps and continuous maps is ultimately due to the fact that for any set S, only is guaranteed in general, whereas for preimages, equality always holds.\n\n# Examples\nThe function defined by f(x) = x^2 is continuous, closed, and relatively open, but not (strongly) open.\nThis is because if U = (a, b) is any open interval in f's domain \\R that does contain 0 then where this open interval is an open subset of both \\R and However, if U = (a, b) is any open interval in \\R that contains 0 then which is not an open subset of f's codomain \\R but an open subset of Because the set of all open intervals in \\R is a [basis](https://bliptext.com/articles/basis-topology) for the [Euclidean topology](https://bliptext.com/articles/euclidean-topology) on \\R, this shows that is relatively open but not (strongly) open.\nIf Y has the [discrete topology](https://bliptext.com/articles/discrete-topology) (that is, all subsets are open and closed) then every function f : X \\to Y is both open and closed (but not necessarily continuous).\nFor example, the [floor function](https://bliptext.com/articles/floor-function) from **[\\R](https://bliptext.com/articles/real-number)** to **[\\Z](https://bliptext.com/articles/integer)** is open and closed, but not continuous.\nThis example shows that the image of a [connected space](https://bliptext.com/articles/connected-space) under an open or closed map need not be connected.\nWhenever we have a [product](https://bliptext.com/articles/product-topology) of topological spaces the natural projections are open (as well as continuous).\nSince the projections of [fiber bundles](https://bliptext.com/articles/fiber-bundle) and [covering maps](https://bliptext.com/articles/covering-map) are locally natural projections of products, these are also open maps.\nProjections need not be closed however.\nConsider for instance the projection on the first component; then the set is closed in \\R^2, but is not closed in \\R.\nHowever, for a compact space Y, the projection is closed.\nThis is essentially the [tube lemma](https://bliptext.com/articles/tube-lemma).\nTo every point on the [unit circle](https://bliptext.com/articles/unit-circle) we can associate the [angle](https://bliptext.com/articles/angle) of the positive x-axis with the ray connecting the point with the origin.\nThis function from the unit circle to the half-open [interval](https://bliptext.com/articles/interval-mathematics) [0,2π) is bijective, open, and closed, but not continuous. It shows that the image of a [compact space](https://bliptext.com/articles/compact-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 the [codomain](https://bliptext.com/articles/codomain) is essential.\n\n# Sufficient conditions\nEvery [homeomorphism](https://bliptext.com/articles/homeomorphism) is open, closed, and continuous.\nIn fact, a [bijective](https://bliptext.com/articles/bijective) continuous map is a homeomorphism [if and only if](https://bliptext.com/articles/if-and-only-if) it is open, or equivalently, [if and only if](https://bliptext.com/articles/if-and-only-if) it is closed.\nThe [composition](https://bliptext.com/articles/function-composition) of two (strongly) open maps is an open map and the [composition](https://bliptext.com/articles/function-composition) of two (strongly) closed maps is a closed map.\nHowever, 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.\nIf f : X \\to Y is strongly open (respectively, strongly closed) and g : Y \\to Z is relatively open (respectively, relatively closed) then is relatively open (respectively, relatively closed).\nLet f : X \\to Y be a map.\nGiven any subset if f : X \\to Y is a relatively open (respectively, relatively closed, strongly open, strongly closed, continuous, [surjective](https://bliptext.com/articles/surjective-function)) map then the same is true of its restriction\nto the [f-saturated](https://bliptext.com/articles/saturated-set) subset f^{-1}(T).\nThe categorical sum of two open maps is open, or of two closed maps is closed.\nThe categorical [product](https://bliptext.com/articles/product-topology) of two open maps is open, however, the categorical [product](https://bliptext.com/articles/product-topology) of two closed maps need not be closed.\nA bijective map is open if and only if it is closed.\nThe inverse of a bijective continuous map is a bijective open/closed map (and vice versa).\nA surjective open map is not necessarily a closed map, and likewise, a surjective closed map is not necessarily an open map.\nAll [local homeomorphisms](https://bliptext.com/articles/local-homeomorphism), including all [coordinate charts](https://bliptext.com/articles/coordinate-chart) on [manifolds](https://bliptext.com/articles/manifold) and all [covering maps](https://bliptext.com/articles/covering-map), are open maps.\nA variant of the closed map lemma states that if a continuous function between [locally compact](https://bliptext.com/articles/locally-compact-space) Hausdorff spaces is proper then it is also closed.\nIn [complex analysis](https://bliptext.com/articles/complex-analysis), the identically named [open mapping theorem](https://bliptext.com/articles/open-mapping-theorem-complex-analysis) states that every non-constant [holomorphic function](https://bliptext.com/articles/holomorphic-function) defined on a [connected](https://bliptext.com/articles/connected-space) open subset of the [complex plane](https://bliptext.com/articles/complex-plane) is an open map.\nThe [invariance of domain](https://bliptext.com/articles/invariance-of-domain) theorem states that a continuous and locally injective function between two n-dimensional [topological manifolds](https://bliptext.com/articles/manifold) must be open.\nIn [functional analysis](https://bliptext.com/articles/functional-analysis), the [open mapping theorem](https://bliptext.com/articles/open-mapping-theorem-functional-analysis) states that every surjective continuous [linear operator](https://bliptext.com/articles/linear-operator) between [Banach spaces](https://bliptext.com/articles/banach-space) is an open map.\nThis theorem has been generalized to [topological vector spaces](https://bliptext.com/articles/topological-vector-space) beyond just Banach spaces.\nA surjective map f : X \\to Y is called an **' if for every y \\in Y there exists some such that x is a **' for f, which by definition means that for every open [neighborhood](https://bliptext.com/articles/neighborhood-topology) U of x, f(U) is a [neighborhood](https://bliptext.com/articles/neighborhood-topology) of f(x) in Y (note that the [neighborhood](https://bliptext.com/articles/neighborhood-topology) f(U) is not required to be an [neighborhood](https://bliptext.com/articles/neighborhood-topology)).\nEvery surjective open map is an almost open map but in general, the converse is not necessarily true.\nIf 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 Y's topology \\sigma):\nIf the map is continuous then the above condition is also necessary for the map to be open.\nThat is, if 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.\n\n# Properties\n\n## Open or closed maps that are continuous\nIf f : X \\to Y is a continuous map that is also open closed then:\nIn the first two cases, being open or closed is merely a [sufficient condition](https://bliptext.com/articles/sufficient-condition) for the conclusion that follows.\nIn the third case, it is [necessary](https://bliptext.com/articles/necessary-condition) as well.\n\n## Open continuous maps\nIf f : X \\to Y is a continuous (strongly) open map, and then:\n

where denotes the [boundary](https://bliptext.com/articles/boundary-topology) of a set.
where denote the [closure](https://bliptext.com/articles/closure-topology) of a set.
If where denotes the [interior](https://bliptext.com/articles/interior-topology) of a set, then\nwhere this set is also necessarily a [regular closed set](https://bliptext.com/articles/regular-closed-set) (in Y).\nIn particular, if A is a regular closed set then so is And if A is a [regular open set](https://bliptext.com/articles/regular-open-set) then so is
If the continuous open map f : X \\to Y is also surjective then and moreover, S is a regular open (resp. a regular closed) subset of Y if and only if f^{-1}(S) is a regular open (resp. a regular closed) subset of X.
If a [net](https://bliptext.com/articles/net-mathematics) [converges](https://bliptext.com/articles/convergent-net) in Y to a point y \\in Y and if the continuous open map f : X \\to Y is surjective, then for any there exists a [net](https://bliptext.com/articles/net-mathematics) in X (indexed by some [directed set](https://bliptext.com/articles/directed-set) A) such that in X and is a sub[net](https://bliptext.com/articles/net-mathematics) of y_{\\bull}.\nMoreover, the indexing set A may be taken to be with the [product order](https://bliptext.com/articles/product-order) where is any [neighbourhood basis](https://bliptext.com/articles/neighbourhood-basis) of x directed by

\n\n# Citations","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T19:21:53.517Z","updated_at":"2025-01-25T19:21:53.517Z","search_vector":"'close':3 'map':4 'open':1","search_content":" \n \n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), more specifically in [topology](https://bliptext.com/articles/topology), an **open map** is a [function](https://bliptext.com/articles/function-mathematics) between two [topological spaces](https://bliptext.com/articles/topological-space) that maps [open sets](https://bliptext.com/articles/open-set) to [open sets](https://bliptext.com/articles/open-set).\nThat is, a function f : X \\to Y is open if for any open set U in X, the [image](https://bliptext.com/articles/image-mathematics) f(U) is open in Y. Likewise, a **closed map** is a function that maps [closed sets](https://bliptext.com/articles/closed-set) to [closed sets](https://bliptext.com/articles/closed-set).\nA map may be open, closed, both, or neither; in particular, an open map need not be closed and vice versa.\nOpen and closed maps are not necessarily [continuous](https://bliptext.com/articles/continuous-function-topology).\nFurther, continuity is independent of openness and closedness in the general case and a continuous function may have one, both, or neither property; this fact remains true even if one restricts oneself to metric spaces.\nAlthough their definitions seem more natural, open and closed maps are much less important than continuous maps.\nRecall that, by definition, a function f : X \\to Y is continuous if the [preimage](https://bliptext.com/articles/preimage) of every open set of Y is open in X. (Equivalently, if the [preimage](https://bliptext.com/articles/preimage) of every closed set of Y is closed in X).\nEarly study of open maps was pioneered by [Simion Stoilow](https://bliptext.com/articles/simion-stoilow) and [Gordon Thomas Whyburn](https://bliptext.com/articles/gordon-thomas-whyburn).\n\n# Definitions and characterizations\nIf S is a subset of a topological space then let and (resp. ) denote the [closure](https://bliptext.com/articles/closure-topology) (resp. [interior](https://bliptext.com/articles/interior-topology)) of S in that space.\nLet f : X \\to Y be a function between [topological spaces](https://bliptext.com/articles/topological-space).\nIf S is any set then is called the image of S under f.\n\n## Competing definitions\nThere are two different competing, but closely related, definitions of \"\" that are widely used, where both of these definitions can be summarized as: \"it is a map that sends open sets to open sets.\"\nThe following terminology is sometimes used to distinguish between the two definitions.\nA map f : X \\to Y is called a\nEvery strongly open map is a relatively open map.\nHowever, these definitions are not equivalent in general.\nA [surjective](https://bliptext.com/articles/surjective-function) map is relatively open if and only if it is strongly open; so for this important special case the definitions are equivalent.\nMore generally, a map f : X \\to Y is relatively open if and only if the [surjection](https://bliptext.com/articles/surjective-function) is a strongly open map.\nBecause X is always an open subset of X, the image of a strongly open map f : X \\to Y must be an open subset of its codomain 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.\nIn summary,\nBy 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.\nThe discussion above will also apply to closed maps if each instance of the word \"open\" is replaced with the word \"closed\".\n\n## Open maps\nA map f : X \\to Y is called an **' or a **' if it satisfies any of the following equivalent conditions:\n \n Definition: f : X \\to Y maps open subsets of its domain to open subsets of its codomain; that is, for any open subset U of X, f(U) is an open subset of Y. \n f : X \\to Y is a relatively open map and its image is an open subset of its codomain Y. \n For every x \\in X and every [neighborhood](https://bliptext.com/articles/neighborhood-topology) N of x (however small), f(N) is a [neighborhood](https://bliptext.com/articles/neighborhood-topology) of f(x).\nWe 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:\n for all subsets A of X, where denotes the [topological interior](https://bliptext.com/articles/topological-interior) of the set. \n Whenever C is a [closed subset](https://bliptext.com/articles/closed-set) of X then the set is a [closed subset](https://bliptext.com/articles/closed-set) of Y. \nIf \\mathcal{B} is a [basis](https://bliptext.com/articles/base-topology) for X then the following can be appended to this list:\n\n## Closed maps\n****A**** ****m****a****p**** ****f**** ****:**** ****X**** ****\\****t****o**** ****Y**** ****i****s**** ****c****a****l****l****e****d**** ****a**** **** ****i****f**** ****w****h****e****n****e****v****e****r**** ****C**** ****i****s**** ****a**** ****c****l****o****s****e****d**** ****s****u****b****s****e****t**** ****o****f**** ****t****h****e**** ****d****o****m****a****i****n**** ****X**** ****t****h****e****n**** ****f****(****C****)**** ****i****s**** ****a**** ****c****l****o****s****e****d**** ****s****u****b****s****e****t**** ****o****f**** ****f****'****s**** ****i****m****a****g****e**** ****w****h****e****r****e**** ****a****s**** ****u****s****u****a****l****,**** ****t****h****i****s**** ****s****e****t**** ****i****s**** ****e****n****d****o****w****e****d**** ****w****i****t****h**** ****t****h****e**** ****s****u****b****s****p****a****c****e**** ****t****o****p****o****l****o****g****y**** ****i****n****d****u****c****e****d**** ****o****n**** ****i****t**** ****b****y**** ****f****'****s**** ****c****o****d****o****m****a****i****n**** ****Y****.****\nA map f : X \\to Y is called a **' or a **' if it satisfies any of the following equivalent conditions:\n \n Definition: f : X \\to Y maps closed subsets of its domain to closed subsets of its codomain; that is, for any closed subset C of X, f(C) is a closed subset of Y. f : X \\to Y is a relatively closed map and its image is a closed subset of its codomain Y. \n for every subset \n for every closed subset \n for every closed subset \n Whenever U is an open subset of X then the set is an open subset of Y. \n If x_{\\bull} is a [net](https://bliptext.com/articles/net-mathematics) in X and y \\in Y is a point such that in Y, then x_{\\bull} converges in X to the set f^{-1}(y).\n \nA [surjective](https://bliptext.com/articles/surjective-function) map is strongly closed if and only if it is relatively closed.\nSo for this important special case, the two definitions are equivalent.\nBy definition, the map f : X \\to Y is a relatively closed map if and only if the [surjection](https://bliptext.com/articles/surjective-function) is a strongly closed map.\nIf in the open set definition of \"[continuous map](https://bliptext.com/articles/continuous-function)\" (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\") is to continuity.\nThis 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 general equivalent to openness.\nThere exist open maps that are not closed and there also exist closed maps that are not open.\nThis difference between open/closed maps and continuous maps is ultimately due to the fact that for any set S, only is guaranteed in general, whereas for preimages, equality always holds.\n\n# Examples\nThe function defined by f(x) = x^2 is continuous, closed, and relatively open, but not (strongly) open.\nThis is because if U = (a, b) is any open interval in f's domain \\R that does contain 0 then where this open interval is an open subset of both \\R and However, if U = (a, b) is any open interval in \\R that contains 0 then which is not an open subset of f's codomain \\R but an open subset of Because the set of all open intervals in \\R is a [basis](https://bliptext.com/articles/basis-topology) for the [Euclidean topology](https://bliptext.com/articles/euclidean-topology) on \\R, this shows that is relatively open but not (strongly) open.\nIf Y has the [discrete topology](https://bliptext.com/articles/discrete-topology) (that is, all subsets are open and closed) then every function f : X \\to Y is both open and closed (but not necessarily continuous).\nFor example, the [floor function](https://bliptext.com/articles/floor-function) from **[\\R](https://bliptext.com/articles/real-number)** to **[\\Z](https://bliptext.com/articles/integer)** is open and closed, but not continuous.\nThis example shows that the image of a [connected space](https://bliptext.com/articles/connected-space) under an open or closed map need not be connected.\nWhenever we have a [product](https://bliptext.com/articles/product-topology) of topological spaces the natural projections are open (as well as continuous).\nSince the projections of [fiber bundles](https://bliptext.com/articles/fiber-bundle) and [covering maps](https://bliptext.com/articles/covering-map) are locally natural projections of products, these are also open maps.\nProjections need not be closed however.\nConsider for instance the projection on the first component; then the set is closed in \\R^2, but is not closed in \\R.\nHowever, for a compact space Y, the projection is closed.\nThis is essentially the [tube lemma](https://bliptext.com/articles/tube-lemma).\nTo every point on the [unit circle](https://bliptext.com/articles/unit-circle) we can associate the [angle](https://bliptext.com/articles/angle) of the positive x-axis with the ray connecting the point with the origin.\nThis function from the unit circle to the half-open [interval](https://bliptext.com/articles/interval-mathematics) [0,2π) is bijective, open, and closed, but not continuous. It shows that the image of a [compact space](https://bliptext.com/articles/compact-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 the [codomain](https://bliptext.com/articles/codomain) is essential.\n\n# Sufficient conditions\nEvery [homeomorphism](https://bliptext.com/articles/homeomorphism) is open, closed, and continuous.\nIn fact, a [bijective](https://bliptext.com/articles/bijective) continuous map is a homeomorphism [if and only if](https://bliptext.com/articles/if-and-only-if) it is open, or equivalently, [if and only if](https://bliptext.com/articles/if-and-only-if) it is closed.\nThe [composition](https://bliptext.com/articles/function-composition) of two (strongly) open maps is an open map and the [composition](https://bliptext.com/articles/function-composition) of two (strongly) closed maps is a closed map.\nHowever, 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.\nIf f : X \\to Y is strongly open (respectively, strongly closed) and g : Y \\to Z is relatively open (respectively, relatively closed) then is relatively open (respectively, relatively closed).\nLet f : X \\to Y be a map.\nGiven any subset if f : X \\to Y is a relatively open (respectively, relatively closed, strongly open, strongly closed, continuous, [surjective](https://bliptext.com/articles/surjective-function)) map then the same is true of its restriction\nto the [f-saturated](https://bliptext.com/articles/saturated-set) subset f^{-1}(T).\nThe categorical sum of two open maps is open, or of two closed maps is closed.\nThe categorical [product](https://bliptext.com/articles/product-topology) of two open maps is open, however, the categorical [product](https://bliptext.com/articles/product-topology) of two closed maps need not be closed.\nA bijective map is open if and only if it is closed.\nThe inverse of a bijective continuous map is a bijective open/closed map (and vice versa).\nA surjective open map is not necessarily a closed map, and likewise, a surjective closed map is not necessarily an open map.\nAll [local homeomorphisms](https://bliptext.com/articles/local-homeomorphism), including all [coordinate charts](https://bliptext.com/articles/coordinate-chart) on [manifolds](https://bliptext.com/articles/manifold) and all [covering maps](https://bliptext.com/articles/covering-map), are open maps.\nA variant of the closed map lemma states that if a continuous function between [locally compact](https://bliptext.com/articles/locally-compact-space) Hausdorff spaces is proper then it is also closed.\nIn [complex analysis](https://bliptext.com/articles/complex-analysis), the identically named [open mapping theorem](https://bliptext.com/articles/open-mapping-theorem-complex-analysis) states that every non-constant [holomorphic function](https://bliptext.com/articles/holomorphic-function) defined on a [connected](https://bliptext.com/articles/connected-space) open subset of the [complex plane](https://bliptext.com/articles/complex-plane) is an open map.\nThe [invariance of domain](https://bliptext.com/articles/invariance-of-domain) theorem states that a continuous and locally injective function between two n-dimensional [topological manifolds](https://bliptext.com/articles/manifold) must be open.\nIn [functional analysis](https://bliptext.com/articles/functional-analysis), the [open mapping theorem](https://bliptext.com/articles/open-mapping-theorem-functional-analysis) states that every surjective continuous [linear operator](https://bliptext.com/articles/linear-operator) between [Banach spaces](https://bliptext.com/articles/banach-space) is an open map.\nThis theorem has been generalized to [topological vector spaces](https://bliptext.com/articles/topological-vector-space) beyond just Banach spaces.\nA surjective map f : X \\to Y is called an **' if for every y \\in Y there exists some such that x is a **' for f, which by definition means that for every open [neighborhood](https://bliptext.com/articles/neighborhood-topology) U of x, f(U) is a [neighborhood](https://bliptext.com/articles/neighborhood-topology) of f(x) in Y (note that the [neighborhood](https://bliptext.com/articles/neighborhood-topology) f(U) is not required to be an [neighborhood](https://bliptext.com/articles/neighborhood-topology)).\nEvery surjective open map is an almost open map but in general, the converse is not necessarily true.\nIf 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 Y's topology \\sigma):\nIf the map is continuous then the above condition is also necessary for the map to be open.\nThat is, if 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.\n\n# Properties\n\n## Open or closed maps that are continuous\nIf f : X \\to Y is a continuous map that is also open closed then:\nIn the first two cases, being open or closed is merely a [sufficient condition](https://bliptext.com/articles/sufficient-condition) for the conclusion that follows.\nIn the third case, it is [necessary](https://bliptext.com/articles/necessary-condition) as well.\n\n## Open continuous maps\nIf f : X \\to Y is a continuous (strongly) open map, and then:\n \n where denotes the [boundary](https://bliptext.com/articles/boundary-topology) of a set. \n where denote the [closure](https://bliptext.com/articles/closure-topology) of a set. \n If where denotes the [interior](https://bliptext.com/articles/interior-topology) of a set, then\nwhere this set is also necessarily a [regular closed set](https://bliptext.com/articles/regular-closed-set) (in Y).\nIn particular, if A is a regular closed set then so is And if A is a [regular open set](https://bliptext.com/articles/regular-open-set) then so is \n If the continuous open map f : X \\to Y is also surjective then and moreover, S is a regular open (resp. a regular closed) subset of Y if and only if f^{-1}(S) is a regular open (resp. a regular closed) subset of X. \n If a [net](https://bliptext.com/articles/net-mathematics) [converges](https://bliptext.com/articles/convergent-net) in Y to a point y \\in Y and if the continuous open map f : X \\to Y is surjective, then for any there exists a [net](https://bliptext.com/articles/net-mathematics) in X (indexed by some [directed set](https://bliptext.com/articles/directed-set) A) such that in X and is a sub[net](https://bliptext.com/articles/net-mathematics) of y_{\\bull}.\nMoreover, the indexing set A may be taken to be with the [product order](https://bliptext.com/articles/product-order) where is any [neighbourhood basis](https://bliptext.com/articles/neighbourhood-basis) of x directed by \n \n\n# Citations","revisionCount":0},"hasMore":false,"nextCursor":null}

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