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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, more specifically in
topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

topology
, an open map is a
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
between two
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
s that maps
open set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
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 (from la, imago) is an artifact that depicts visual perception Visual perception is the ability to interpret the surrounding environment (biophysical), environment through photopic vision (daytime vision), color vision, sco ...
f(U) is open in Y. Likewise, a closed map is a function that maps
closed set In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...
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 Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ga ...
. 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
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 Simion Stoilow or Stoilov ( – 4 April 1961) was a Romania Romania ( ; ro, România ) is a country at the crossroads of Central Europe, Central, Eastern Europe, Eastern and Southeast Europe, Southeastern Europe. It borders Bulgaria to the s ...
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 Interior may refer to: Arts and media * Interior (Degas), ''Interior'' (Degas) (also known as ''The Rape''), painting by Edgar Degas * Interior (play), ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck * The Interior (novel) ...
) of S in that space. Let f : X \to Y be a function between
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
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 Open or OPEN may refer to: Music * Open (band) Open is a band. Background Drummer Pete Neville has been involved in the Sydney/Australian music scene for a number of years. He has recently completed a Masters in screen music at the Australian ...
of the domain X then f(U) is an open subset of f's
codomain In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

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 (from la, imago) is an artifact that depicts visual perception Visual perception is the ability to interpret the surrounding environment (biophysical), environment through photopic vision (daytime vision), color vision, sco ...
\operatorname f := f(X), where as usual, this set is endowed with the
subspace topology In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and ...

subspace topology
induced on it by f's codomain Y. A
surjective In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

surjective
map is relatively open if and only if it strongly open; so for this important special case the definitions are equivalent. More generally, the map f : X \to Y is a relatively open map if and only if the
surjection In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

surjection
f : X \to \operatorname f is a strongly open map. :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. Every strongly open map is a relatively open map. And 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 Y. However, a relatively open map f : X \to Y is a strongly open map if and only if its image \operatorname f is an open subset of its codomain Y. 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 (British English British English (BrE) is the standard dialect A standard language (also standard variety, standard dialect, and standard) is a language variety that has undergone substantial codification of grammar ...
    N of x (however small), there exists a neighborhood V of f(x) such that V \subseteq f(N). * 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 In mathematics, specifically in general topology, topology, the interior of a subset of a topological space is the Union (set theory), union of all subsets of that are Open set, open in . A point that is in the interior of is an interior point ...
    of the set.
  5. Whenever C is a
    closed subset In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...
    of X then the set \left\ is a closed subset of Y. * This is a consequence of the
    identity Identity may refer to: Social sciences * Identity (social science), personhood or group affiliation in psychology and sociology Group expression and affiliation * Cultural identity, a person's self-affiliation (or categorization by others ...
    f(X \setminus R) = Y \setminus \left\, which holds for all subsets R \subseteq X.
and if \mathcal is a
basis Basis may refer to: Finance and accounting *Adjusted basisIn tax accounting, adjusted basis is the net cost of an asset after adjusting for various tax-related items. Adjusted Basis or Adjusted Tax Basis refers to the original cost or other b ...
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 In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...
    of the domain X then f(C) is a closed subset of f's
    image An image (from la, imago) is an artifact that depicts visual perception Visual perception is the ability to interpret the surrounding environment (biophysical), environment through photopic vision (daytime vision), color vision, sco ...
    \operatorname f := f(X), where as usual, this set is endowed with the
    subspace topology In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and ...

    subspace topology
    induced on it by f's
    codomain In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

    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. Whenever U is an open subset of X then the set \left\ is an open subset of Y.
    A
    surjective In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

    surjective
    map is strongly closed if and only if it a 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

    surjection
    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 (mathematics), function such that a continuous variation (that is a change without jump) of the argument of a function, argument induces a continuous variation of the Value (mathematics), value o ...
    " (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 basisIn tax accounting, adjusted basis is the net cost of an asset after adjusting for various tax-related items. Adjusted Basis or Adjusted Tax Basis refers to the original cost or other b ...
    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 (mathematics), 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 ...

    floor function
    from \R to \Z is open and closed, but not continuous. This example shows that the image of a
    connected space In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry) ...
    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 bundle In mathematics, and particularly topology, a fiber bundle (or, in English in the Commonwealth of Nations, Commonwealth English: fibre bundle) is a Space (mathematics), space that is ''locally'' a product space, but ''globally'' may have a dif ...
    s and
    covering map In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

    covering map
    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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no genera ...
    . To every point on the
    unit circle In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

    unit circle
    we can associate the
    angle In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method ...

    angle
    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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

    codomain
    is essential.


    Sufficient conditions

    Every
    homeomorphism In the mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantiti ...
    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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

    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 homeomorphism In mathematics, more specifically topology, a local homeomorphism is a Function (mathematics), function between topological spaces that, intuitively, preserves local (though not necessarily global) structure. If f : X \to Y is a local homeomorphism ...
    s, including all
    coordinate chartIn topology, a branch of mathematics, a topological manifold is a topological space which locally resembles real numbers, real ''n''-dimension (mathematics), dimensional Euclidean space. Topological manifolds are an important class of topological spa ...
    s on
    manifold In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

    manifold
    s and all
    covering map In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

    covering map
    s, are open maps. A variant of the closed map lemma states that if a continuous function between
    locally compact In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
    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 Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Der ...
    , the identically named open mapping theorem states that every non-constant
    holomorphic function In mathematics, a holomorphic function is a complex-valued function Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), ...
    defined on a connected open subset of the
    complex plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
    is an open map. The invariance of domain theorem states that a continuous and locally injective function between two n-dimensional must be open. In
    functional analysis Functional analysis is a branch of mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), ...
    , the open mapping theorem states that every surjective continuous
    linear operator In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
    between
    Banach space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
    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 (an Abstra ...
    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 (British English British English (BrE) is the standard dialect A standard language (also standard variety, standard dialect, and standard) is a language variety that has undergone substantial codification of grammar ...
    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 Nature, in the broadest sense, is the natural, physical, material world or universe The universe ( la, universus) is all of space and time and their contents, including ...
    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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), ...
    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 * Injection (medicine) An injection (often referred to as a "shot" in US English, a "jab" in UK English, or a "jag" in Scottish English and Scots Language, Scots) is the act of adminis ...

    injection
    then it is a
    topological embedding In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
    . * if f is a
    bijection In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

    bijection
    then it is a
    homeomorphism In the mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantiti ...
    . In the first two cases, being open or closed is merely a
    sufficient condition In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, ac ...
    for the conclusion that follows. In the third case, it is
    necessary Necessary or necessity may refer to: * Need A need is something that is necessary Necessary or necessity may refer to: * Need ** An action somebody may feel they must do ** An important task or essential thing to do at a particular time or by ...
    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), ''Boundaries'' (2016 film), a 2016 Canadian film *Boundaries (2018 film), ''Boundaries'' (2018 film), a 2018 American-Canadian road trip ...
      of a set.
    • f^\left(\overline\right) = \overline where \overline denote the closure of a set.
    • If \overline = \overline, where \operatorname denotes the
      interior Interior may refer to: Arts and media * Interior (Degas), ''Interior'' (Degas) (also known as ''The Rape''), painting by Edgar Degas * Interior (play), ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck * The Interior (novel) ...
      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 setA semiregular space is a topological space whose regular open sets (sets that equal the interiors of their closures) form a Base (topology), base. Semiregular spaces should not be confused with locally regular spaces, spaces in which there is a base ...
      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) In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they ...
      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(y_a\right)_ in X (indexed by some
      directed set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
      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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
      where \mathcal_x is any
      neighbourhood basisIn topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter \mathcal(x) for a point is the collection of all Neighbourhood (mathematics), neighbourhoods of the point . Definit ...
      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, let ''A'' be a set and let be a binary relation on ''A''. Then a subset is said to be cofinal or frequent in ''A'' if it satisfies the following condition: :For every , there exists some such that . A subset that is not frequ ...
      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 Continuous mappings Lemmas