In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, more specifically in
topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
, 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 points ...
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 suf ...
s to open sets.
That is, a function
is open if for any open set
in
the
image
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
is open in
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 whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a cl ...
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 ...
.[ 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 is continuous if the ]preimage
In mathematics, the image of a function is the set of all output values it may produce.
More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or through) ...
of every open set of is open in [ (Equivalently, if the preimage of every closed set of is closed in ).
Early study of open maps was pioneered by ]Simion Stoilow
Simion Stoilow or Stoilov ( – 4 April 1961) was a Romanian mathematician, creator of the Romanian school of complex analysis, and author of over 100 publications.
Biography
He was born in Bucharest, and grew up in Craiova. His father, Colonel ...
and Gordon Thomas Whyburn
Gordon Thomas Whyburn (7 January 1904 Lewisville, Texas – 8 September 1969 Charlottesville, Virginia) was an American mathematician who worked on topology.
Whyburn studied at the University of Texas in Austin, where he received a bachel ...
.
Definitions and characterizations
If is a subset of a topological space then let and (resp. ) denote the closure (resp. interior
Interior may refer to:
Arts and media
* ''Interior'' (Degas) (also known as ''The Rape''), painting by Edgar Degas
* ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck
* ''The Interior'' (novel), by Lisa See
* Interior de ...
) of in that space.
Let 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 points ...
s. If is any set then is called the image of under
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 is called a
* "" if whenever is an open subset
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 suff ...
of the domain then is an open subset of '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 the ...
* "" if whenever is an open subset of the domain then is an open subset of 's image
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
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 's codomain
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
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 i ...
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 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 i ...
is a strongly open map.
Because is always an open subset of the image of a strongly open map must be an open subset of its codomain 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 is called an or a if it satisfies any of the following equivalent conditions:
- Definition: maps open subsets of its domain to open subsets of its codomain; that is, for any open subset of , is an open subset of
- is a relatively open map and its image is an open subset of its codomain
- For every and every
neighborhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
of (however small), is a neighborhood of .
* 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.
- for all subsets of where denotes the
topological interior
In mathematics, specifically in topology,
the interior of a subset of a topological space is the union of all subsets of that are open in .
A point that is in the interior of is an interior point of .
The interior of is the complement of the ...
of the set.
- Whenever is a
closed subset
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a clo ...
of then the set is a closed subset of
* This is a consequence of the identity
Identity may refer to:
* Identity document
* Identity (philosophy)
* Identity (social science)
* Identity (mathematics)
Arts and entertainment Film and television
* ''Identity'' (1987 film), an Iranian film
* ''Identity'' (2003 film), ...
which holds for all subsets
If 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 of ...
for then the following can be appended to this list:
# maps basic open sets to open sets in its codomain (that is, for any basic open set is an open subset of ).
Closed maps
A map is called a if whenever is a closed subset
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a clo ...
of the domain then is a closed subset of 's image
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
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 '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 the ...
A map is called a or a if it satisfies any of the following equivalent conditions:
- Definition: maps closed subsets of its domain to closed subsets of its codomain; that is, for any closed subset of is a closed subset of
- is a relatively closed map and its image is a closed subset of its codomain
- for every subset
- for every closed subset
- for every closed subset
- Whenever is an open subset of then the set is an open subset of
- If 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 and is a point such that in then converges in to the set
* The convergence means that every open subset of that contains will contain for all sufficiently large indices
A surjective
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 i ...
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 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 i ...
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 value ...
" (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 only is guaranteed in general, whereas for preimages, equality always holds.
Examples
The function defined by is continuous, closed, and relatively open, but not (strongly) open. This is because if is any open interval in 's domain that does contain then where this open interval is an open subset of both and -saturated">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 é ...
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 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 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 sp ...
s, are open maps.
A variant of the closed map lemma states that if a continuous function between locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...
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 Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
, 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 derivativ ...
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
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
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 space#Definition, inner product, Norm (mathematics)#Defini ...
, 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 pre ...
between Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) 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 vector ...
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 (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
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 incorporate ...
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
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
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 isomorphi ...
.
In the first two cases, being open or closed is merely a sufficient condition
In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth of ...
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), ''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) (also known as ''The Rape''), painting by Edgar Degas
* ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck
* ''The Interior'' (novel), by Lisa See
* Interior de ...
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 has ...
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 ...
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
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of orde ...
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).
Cofin ...
of I; thus f\left(x_\right) is a Willard-subnet In topology and related areas of mathematics, a subnet is a generalization of the concept of subsequence to the case of nets. The analogue of "subsequence" for nets is the notion of a "subnet". The definition is not completely straightforward, but ...
of y_.
See also
*
*
*
*
*
*
*
*
*
Notes
Citations
References
*
*
*
{{DEFAULTSORT:Open And Closed Maps
General topology
Theory of continuous functions
Lemmas