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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, more specifically in
topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
, an open map is a function between two
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
s that maps
open set In mathematics, an open set is a generalization of an Interval (mathematics)#Definitions_and_terminology, open interval in the real line. In a metric space (a Set (mathematics), set with a metric (mathematics), distance defined between every two ...
s to open sets. That is, a function f : X \to Y is open if for any open set U in X, the
image An image or picture is a visual representation. An image can be Two-dimensional space, two-dimensional, such as a drawing, painting, or photograph, or Three-dimensional space, three-dimensional, such as a carving or sculpture. Images may be di ...
f(U) is open in Y. Likewise, a closed map is a function that maps
closed set In geometry, topology, and related branches of mathematics, a closed set is a Set (mathematics), set whose complement (set theory), complement is an open set. In a topological space, a closed set can be defined as a set which contains all its lim ...
s 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 Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
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 Y. * "" if whenever U is an open subset of the domain X then f(U) is an open subset of f's
image An image or picture is a visual representation. An image can be Two-dimensional space, two-dimensional, such as a drawing, painting, or photograph, or Three-dimensional space, three-dimensional, such as a carving or sculpture. Images may be di ...
\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'', ''𝜏'') is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''𝜏'' called the subspace topology (or the relative topology ...
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 is 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 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 A neighbourhood (Commonwealth English) or neighborhood (American English) is a geographically localized community within a larger town, city, suburb or rural area, sometimes consisting of a single street and the buildings lining it. Neigh ...
    N of x (however small), f(N) is a neighborhood of 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 x \in X and every open neighborhood N of x, f(N) is a neighborhood of f(x). * For every x \in X and every open neighborhood N of x, f(N) is an open neighborhood of f(x).
  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 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 An image or picture is a visual representation. An image can be Two-dimensional space, two-dimensional, such as a drawing, painting, or photograph, or Three-dimensional space, three-dimensional, such as a carving or sculpture. Images may be di ...
    \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'', ''𝜏'') is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''𝜏'' called the subspace topology (or the relative topology ...
    induced on it by f's codomain 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 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 f : X \to \operatorname 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") 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 for the Euclidean topology">, \infty). Because the set of all open intervals in \R is a Basis (topology)">basis 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 from Real number, \R to Integer, \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 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 maps 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. 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 can refer to a number of concepts relating to the intersection of two straight Line (geometry), lines at a Point (geometry), point. Formally, an angle is a figure lying in a Euclidean plane, plane formed by two R ...
    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 is essential.


    Sufficient conditions

    Every
    homeomorphism In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function ...
    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 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 homeomorphisms, including all coordinate charts on
    manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...
    s and all covering maps, 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 Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...
    , the identically named open mapping theorem states that every non-constant
    holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex de ...
    defined on a connected open subset of the complex plane is an open map. The invariance of domain 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 (for example, Inner product space#Definition, inner product, Norm (mathematics ...
    , the open mapping theorem states that every surjective continuous
    linear operator In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
    between
    Banach space In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...
    s 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 A neighbourhood (Commonwealth English) or neighborhood (American English) is a geographically localized community within a larger town, city, suburb or rural area, sometimes consisting of a single street and the buildings lining it. Neigh ...
    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 (spelled fibre in British English; from ) 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 inco ...
    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 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 then it is a topological embedding. * if f is a
    bijection In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...
    then it is a
    homeomorphism In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function ...
    . 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 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 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 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 preordered set in which every finite subset has an upper bound. In other words, it is a non-empty preordered set A such that for any a and b in A there exists c in A wit ...
      A) such that x_ \to x in X and f\left(x_\right) := \left(f\left(x_a\right)\right)_ is a subnet of y_. Moreover, the indexing set A may be taken to be A := I \times \mathcal_x with the
      product order In mathematics, given partial orders \preceq and \sqsubseteq on sets A and B, respectively, the product order (also called the coordinatewise order or componentwise order) is a partial order \leq on the Cartesian product A \times B. Given two pa ...
      where \mathcal_x is any neighbourhood basis 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 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