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In mathematics, more specifically in
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, an open map is a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
between two
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
s that maps
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
s to open sets. That is, a function f : X \to Y is open if for any open set U in X, the image f(U) is open in Y. Likewise, a closed map is a function that maps closed sets to closed sets. A map may be open, closed, both, or neither; in particular, an open map need not be closed and vice versa. Open and closed maps are not necessarily continuous. Further, 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. Although their definitions seem more natural, open and closed maps are much less important than continuous maps. Recall that, by definition, a function f : X \to Y is continuous if the preimage of every open set of Y is open in X. (Equivalently, if the preimage of every closed set of Y is closed in X). Early study of open maps was pioneered by Simion Stoilow and Gordon Thomas Whyburn.


Definitions and characterizations

If S is a subset of a topological space then let \overline and \operatorname S (resp. \operatorname S) denote the closure (resp. interior) of S in that space. Let f : X \to Y be a function between
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
s. If S is any set then f(S) := \left\ is called the image of S under f.


Competing definitions

There 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." The following terminology is sometimes used to distinguish between the two definitions. A map f : X \to Y is called a * "" if whenever U is an open subset of the domain X then f(U) is an open subset of f's
codomain In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation . The term range is sometimes ambiguously used to refer to either th ...
Y. * "" if whenever U is an open subset of the domain X then f(U) is an open subset of f's image \operatorname f := f(X), where as usual, this set is endowed with the
subspace topology In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced to ...
induced on it by f's codomain Y. 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 " open map" (for example, The Encyclopedia of Mathematics) while others define "open map" to mean " open map". In general, these definitions are equivalent so it is thus advisable to always check what definition of "open map" an author is using. A surjective map is relatively open if and only if it strongly open; so for this important special case the definitions are equivalent. More generally, a map f : X \to Y is relatively open if and only if the
surjection In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of ...
f : X \to f(X) is a strongly open map. Because X is always an open subset of X, the image f(X) = \operatorname f 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. 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".


Open maps

A map f : X \to Y is called an or a if it satisfies any of the following equivalent conditions:
  1. 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.
  2. f : X \to Y is a relatively open map and its image \operatorname f := f(X) is an open subset of its codomain Y.
  3. For every x \in X and every neighborhood N of x (however small), f(N) is a neighborhood of f(x). * Either instance of the word "neighborhood" in this statement can be replaced with "open neighborhood" and the resulting statement would still characterize strongly open maps.
  4. f\left( \operatorname_X A \right) \subseteq \operatorname_Y ( f(A) ) for all subsets A of X, where \operatorname denotes the topological interior of the set.
  5. Whenever C is a closed subset of X then the set \left\ is a closed subset of Y. * This is a consequence of the identity f(X \setminus R) = Y \setminus \left\, which holds for all subsets R \subseteq X.
If \mathcal is a
basis Basis may refer to: Finance and accounting * Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates * Basis trading, a trading strategy consisting ...
for X then the following can be appended to this list: #
  • f maps basic open sets to open sets in its codomain (that is, for any basic open set B \in \mathcal, f(B) is an open subset of Y).

  • Closed maps

    A map f : X \to Y is called a if whenever C is a closed subset of the domain X then f(C) is a closed subset of f's image \operatorname f := f(X), where as usual, this set is endowed with the
    subspace topology In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced to ...
    induced on it by f's
    codomain In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation . The term range is sometimes ambiguously used to refer to either th ...
    Y. A map f : X \to Y is called a or a if it satisfies any of the following equivalent conditions:
    1. 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.
    2. f : X \to Y is a relatively closed map and its image \operatorname f := f(X) is a closed subset of its codomain Y.
    3. \overline \subseteq f\left(\overline\right) for every subset A \subseteq X.
    4. \overline \subseteq f(C) for every closed subset C \subseteq X.
    5. \overline = f(C) for every closed subset C \subseteq X.
    6. Whenever U is an open subset of X then the set \left\ is an open subset of Y.
    7. If x_ is a
      net Net or net may refer to: Mathematics and physics * Net (mathematics), a filter-like topological generalization of a sequence * Net, a linear system of divisors of dimension 2 * Net (polyhedron), an arrangement of polygons that can be folded up ...
      in X and y \in Y is a point such that f\left(x_\right) \to y in Y, then x_ converges in X to the set f^(y). * The convergence x_ \to f^(y) means that every open subset of X that contains f^(y) will contain x_j for all sufficiently large indices j.
    A surjective 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 f : X \to Y is a relatively closed map if and only if the
    surjection In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of ...
    f : X \to \operatorname f is a strongly closed map. If in the open set definition of "
    continuous map In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in valu ...
    " (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. 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 general 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 S, only f(X \setminus S) \supseteq f(X) \setminus f(S) is guaranteed in general, whereas for preimages, equality f^(Y \setminus S) = f^(Y) \setminus f^(S) always holds.


    Examples

    The function f : \R \to \R defined by f(x) = x^2 is continuous, closed, and relatively open, but not (strongly) open. This is because if U = (a, b) is any open interval in f's domain \R that does contain 0 then f(U) = (\min \, \max \), where this open interval is an open subset of both \R and \operatorname f := f(\R) =
    basis Basis may refer to: Finance and accounting * Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates * Basis trading, a trading strategy consisting ...
    for the Euclidean topology on \R, this shows that f : \R \to \R is relatively open but not (strongly) open. If Y has the 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). For example, the
    floor function In mathematics and computer science, the floor function is the function that takes as input a real number , and gives as output the greatest integer less than or equal to , denoted or . Similarly, the ceiling function maps to the least int ...
    from \R to \Z is open and closed, but not continuous. This example shows that the image of a connected space under an open or closed map need not be connected. Whenever we have a
    product Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Produ ...
    of topological spaces X=\prod X_i, the natural projections p_i : X \to X_i are open (as well as continuous). Since the projections of fiber bundles and
    covering map A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties. Definition Let X be a topological space. A covering of X is a continuous map : \pi : E \rightarrow X such that there exists a discrete spa ...
    s are locally natural projections of products, these are also open maps. Projections need not be closed however. Consider for instance the projection p_1 : \R^2 \to \R on the first component; then the set A = \ is closed in \R^2, but p_1(A) = \R \setminus \ is not closed in \R. However, for a compact space Y, the projection X \times Y \to X is closed. This is essentially the
    tube lemma In mathematics, particularly topology, the tube lemma is a useful tool in order to prove that the finite product of compact spaces is compact. Statement The lemma uses the following terminology: * If X and Y are topological spaces and X \times ...
    . To every point on the
    unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
    we can associate the
    angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles a ...
    of the positive x-axis with the ray connecting the point with the origin. This function from the unit circle to the half-open interval ,2π) is bijective, open, and closed, but not continuous. It shows that the image of a 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 In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation . The term range is sometimes ambiguously used to refer to either th ...
    is essential.


    Sufficient conditions

    Every
    homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...
    is open, closed, and continuous. In fact, a bijective continuous map is a homeomorphism if and only if it is open, or equivalently, if and only if it is closed. The Function composition, composition of two (strongly) open maps is an open map and the composition of two (strongly) closed maps is a closed map. 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 f : X \to Y is strongly open (respectively, strongly closed) and g : Y \to Z is relatively open (respectively, relatively closed) then g \circ f : X \to Z is relatively open (respectively, relatively closed). Let f : X \to Y be a map. Given any subset T \subseteq Y, if 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 f\big\vert_ ~:~ f^(T) \to T to the Saturated set, f-saturated subset f^(T). The categorical sum of two open maps is open, or of two closed maps is closed. The categorical
    product Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Produ ...
    of two open maps is open, however, the categorical product of two closed maps need not be closed. 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. All
    local homeomorphism In mathematics, more specifically topology, a local homeomorphism is a function between topological spaces that, intuitively, preserves local (though not necessarily global) structure. If f : X \to Y is a local homeomorphism, X is said to be an Ă ...
    s, including all
    coordinate chart In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real ''n''-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathe ...
    s on manifolds and all
    covering map A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties. Definition Let X be a topological space. A covering of X is a continuous map : \pi : E \rightarrow X such that there exists a discrete spa ...
    s, are open maps. A variant of the closed map lemma states that if a continuous function between locally compact Hausdorff spaces is proper then it is also closed. In complex analysis, the identically named open mapping theorem states that every non-constant holomorphic function defined on a
    connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...
    open subset of the complex plane is an open map. The
    invariance of domain Invariance of domain is a theorem in topology about homeomorphic subsets of Euclidean space \R^n. It states: :If U is an open subset of \R^n and f : U \rarr \R^n is an injective continuous map, then V := f(U) is open in \R^n and f is a homeomorph ...
    theorem states that a continuous and locally injective function between two n-dimensional topological manifolds must be open. In
    functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined o ...
    , the open mapping theorem states that every surjective continuous linear operator between Banach spaces is an open map. This theorem has been generalized to
    topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
    s beyond just Banach spaces. A surjective map f : X \to Y is called an if for every y \in Y there exists some x \in f^(y) such that x is a for f, which by definition means that for every open neighborhood U of x, f(U) is a neighborhood of f(x) in Y (note that the neighborhood f(U) is not required to be an neighborhood). Every surjective open map is an almost open map but in general, the converse is not necessarily true. If a surjection 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 depend in any way on Y's topology \sigma): :whenever m, n \in X belong to the same
    fiber Fiber or fibre (from la, fibra, links=no) is a natural or artificial substance that is significantly longer than it is wide. Fibers are often used in the manufacture of other materials. The strongest engineering materials often incorpora ...
    of f (that is, f(m) = f(n)) then for every neighborhood U \in \tau of m, there exists some neighborhood V \in \tau of n such that 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 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.


    Properties


    Open or closed maps that are continuous

    If f : X \to Y is a continuous map that is also open closed then: * if f is a surjection then it is a
    quotient map In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient to ...
    and even a hereditarily quotient map, ** A surjective map f : X \to Y is called if for every subset T \subseteq Y, the restriction f\big\vert_ ~:~ f^(T) \to T is a quotient map. * if f is an
    injection Injection or injected may refer to: Science and technology * Injective function, a mathematical function mapping distinct arguments to distinct values * Injection (medicine), insertion of liquid into the body with a syringe * Injection, in broadca ...
    then it is a
    topological embedding In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup. When some object X is said to be embedded in another object Y, the embedding is giv ...
    . * if f is a bijection then it is a
    homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...
    . In the first two cases, being open or closed is merely a sufficient condition for the conclusion that follows. In the third case, it is necessary as well.


    Open continuous maps

    If f : X \to Y is a continuous (strongly) open map, A \subseteq X, and S \subseteq Y, then:
    • f^\left(\operatorname_Y S\right) = \operatorname_X \left(f^(S)\right) where \operatorname denotes the
      boundary Boundary or Boundaries may refer to: * Border, in political geography Entertainment * ''Boundaries'' (2016 film), a 2016 Canadian film * ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film *Boundary (cricket), the edge of the pla ...
      of a set.
    • f^\left(\overline\right) = \overline where \overline denote the closure of a set.
    • If \overline = \overline, where \operatorname denotes the interior of a set, then \overline = \overline = \overline = \overline where this set \overline is also necessarily a regular closed set (in Y). In particular, if A is a regular closed set then so is \overline. And if A is a
      regular open set A subset S of a topological space X is called a regular open set if it is equal to the interior of its closure; expressed symbolically, if \operatorname(\overline) = S or, equivalently, if \partial(\overline)=\partial S, where \operatorname S, \ov ...
      then so is Y \setminus \overline.
    • If the continuous open map f : X \to Y is also surjective then \operatorname_X f^(S) = f^\left(\operatorname_Y S\right) and moreover, S is a regular open (resp. a regular closed) subset of Y if and only if f^(S) is a regular open (resp. a regular closed) subset of X.
    • If a
      net Net or net may refer to: Mathematics and physics * Net (mathematics), a filter-like topological generalization of a sequence * Net, a linear system of divisors of dimension 2 * Net (polyhedron), an arrangement of polygons that can be folded up ...
      y_ = \left(y_i\right)_ converges in Y to a point y \in Y and if the continuous open map f : X \to Y is surjective, then for any x \in f^(y) there exists a net x_ = \left(x_a\right)_ in X (indexed by some
      directed set In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive and transitive binary relation \,\leq\, (that is, a preorder), with the additional property that every pair of elements ha ...
      A) such that x_ \to x in X and f\left(x_\right) := \left(f\left(x_a\right)\right)_ is a
      subnet A subnetwork or subnet is a logical subdivision of an IP network. Updated by RFC 6918. The practice of dividing a network into two or more networks is called subnetting. Computers that belong to the same subnet are addressed with an identical ...
      of y_. Moreover, the indexing set A may be taken to be A := I \times \mathcal_x with the
      product order In mathematics, given two preordered sets A and B, the product order (also called the coordinatewise orderDavey & Priestley, ''Introduction to Lattices and Order'' (Second Edition), 2002, p. 18 or componentwise order) is a partial ordering o ...
      where \mathcal_x is any
      neighbourhood basis In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter \mathcal(x) for a point x in a topological space is the collection of all neighbourhoods of x. Definitions Neighbour ...
      of x directed by \,\supseteq.\,Explicitly, for any a := (i, U) \in A := I \times \mathcal_x, pick any h_a \in I such that i \leq h_a \text y_ \in f(U) and then let x_a \in U \cap f^\left(y_\right) be arbitrary. The assignment a \mapsto h_a defines an order morphism h : A \to I such that h(A) is a
      cofinal subset In mathematics, a subset B \subseteq A of a preordered set (A, \leq) is said to be cofinal or frequent in A if for every a \in A, it is possible to find an element b in B that is "larger than a" (explicitly, "larger than a" means a \leq b). Cofi ...
      of I; thus f\left(x_\right) is a Willard-subnet of y_.


    See also

    * * * * * * * * *


    Notes


    Citations


    References

    * * * {{DEFAULTSORT:Open And Closed Maps General topology Theory of continuous functions Lemmas