Open and closed maps
Encyclopedia
In topology
, an open map is a function
between two topological space
s which maps open set
s to open sets. That is, a function f : X → 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
which maps closed set
s to closed sets. (The concept of a closed map should not be confused with that of a closed operator
.)
Neither open nor closed maps are required to be continuous. Although their definitions seem more natural, open and closed maps are much less important than continuous maps. Recall that a function f : X → 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).
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.
If Y has the discrete topology (i.e. all subsets are open and closed) then every function f : X → Y is both open and closed (but not necessarily continuous). For example, the floor function
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
of topological spaces X=ΠXi, the natural projections pi : X → Xi are open (as well as continuous).
Since the projections of fiber bundle
s and 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 p1 : R2 → R on the first component; A = {(x,1/x) : x≠0} is closed in R2, but p1(A) = R − {0} is not closed. However, for compact Y, the projection X × Y → X is closed. This is essentially the tube lemma
.
To every point on the unit circle
we can associate the 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
[0,2π) 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.
The function f : R → R with f(x) = x2 is continuous and closed, but not open.
for every x in X and every neighborhood U of x (however small), there exists a neighborhood V of f(x) such that V ⊂ f(U).
It suffices to check openness on a basis
for X. That is, a function f : X → Y is open if and only if it maps basic open sets to open sets.
Open and closed maps can also be characterized by the interior and closure operator
s. Let f : X → Y be a function. Then
The composition
of two open maps is again open; the composition of two closed maps is again closed.
The product of two open maps is open, however the 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.
Let f : X → Y be a continuous map which is either open or closed. Then
In the first two cases, being open or closed is merely a sufficient condition for the result to follow. In the third case it is necessary as well.
The closed map lemma states that every continuous function f : X → Y from a compact space
X to a Hausdorff space
Y is closed and proper
(i.e. preimages of compact sets are compact). A variant of this result states that if a continuous function between locally compact
Hausdorff spaces is proper, then it is also closed.
In functional analysis
, the open mapping theorem
states that every surjective continuous linear operator between Banach space
s is an open map.
In complex analysis
, the identically named open mapping theorem
states that every non-constant holomorphic function
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.
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, an open map is a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
between two topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
s which maps open set
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
s to open sets. That is, a function f : X → Y is open if for any open set U in X, the image
Image (mathematics)
In mathematics, an image is the subset of a function's codomain which is the output of the function on a subset of its domain. Precisely, evaluating the function at each element of a subset X of the domain produces a set called the image of X under or through the function...
f(U) is open in Y. Likewise, a closed map is a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
which maps closed set
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...
s to closed sets. (The concept of a closed map should not be confused with that of a closed operator
Closed operator
In mathematics, specifically in functional analysis, closed linear operators are an important class of linear operators on Banach spaces. They are more general than bounded operators, and therefore not necessarily continuous, but they still retain nice enough properties that one can define the...
.)
Neither open nor closed maps are required to be continuous. Although their definitions seem more natural, open and closed maps are much less important than continuous maps. Recall that a function f : X → 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).
Examples
Every homeomorphismHomeomorphism
In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...
is open, closed, and continuous. In fact, a bijective continuous map is a homeomorphism if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
it is open, or equivalently, if and only if it is closed.
If Y has the discrete topology (i.e. all subsets are open and closed) then every function f : X → Y is both open and closed (but not necessarily continuous). For example, the floor function
Floor function
In mathematics and computer science, the floor and ceiling functions map a real number to the largest previous or the smallest following integer, respectively...
from R
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
to Z
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
is open and closed, but not continuous. This example shows that the image of a connected space
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
under an open or closed map need not be connected.
Whenever we have a product
Product topology
In topology and related areas of mathematics, a product space is the cartesian product of a family of topological spaces equipped with a natural topology called the product topology...
of topological spaces X=ΠXi, the natural projections pi : X → Xi are open (as well as continuous).
Since the projections of fiber bundle
Fiber bundle
In mathematics, and particularly topology, a fiber bundle is intuitively a space which locally "looks" like a certain product space, but globally may have a different topological structure...
s and covering map
Covering map
In mathematics, more specifically algebraic topology, a covering map is a continuous surjective function p from a topological space, C, to a topological space, X, such that each point in X has a neighbourhood evenly covered by p...
s are locally natural projections of products, these are also open maps. Projections need not be closed however. Consider for instance the projection p1 : R2 → R on the first component; A = {(x,1/x) : x≠0} is closed in R2, but p1(A) = R − {0} is not closed. However, for compact Y, the projection X × Y → X is closed. This is essentially the tube lemma
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. It is in general, a concept of point-set topology.-Tube lemma:...
.
To every point on the unit circle
Unit circle
In mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, "the" unit circle is the circle of radius one centered at the origin in the Cartesian coordinate system in the Euclidean plane...
we can associate the angle
Angle
In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...
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
Interval (mathematics)
In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
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
Codomain
In mathematics, the codomain or target set 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...
is essential.
The function f : R → R with f(x) = x2 is continuous and closed, but not open.
Properties
A function f : X → Y is open if and only ifIf and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
for every x in X and every neighborhood U of x (however small), there exists a neighborhood V of f(x) such that V ⊂ f(U).
It suffices to check openness on a basis
Base (topology)
In mathematics, a base B for a topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements of B. We say that the base generates the topology T...
for X. That is, a function f : X → Y is open if and only if it maps basic open sets to open sets.
Open and closed maps can also be characterized by the interior and closure operator
Closure operator
In mathematics, a closure operator on a set S is a function cl: P → P from the power set of S to itself which satisfies the following conditions for all sets X,Y ⊆ S....
s. Let f : X → Y be a function. Then
- f is open if and only if f(A°) ⊆ f(A)° for all A ⊆ X
- f is closed if and only if f(A)− ⊂ f(A−) for all A ⊂ X
The composition
Function composition
In mathematics, function composition is the application of one function to the results of another. For instance, the functions and can be composed by computing the output of g when it has an argument of f instead of x...
of two open maps is again open; the composition of two closed maps is again closed.
The product of two open maps is open, however the 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.
Let f : X → Y be a continuous map which is either open or closed. Then
- if f is a surjection, then it is a quotient map,
- if f is an injection, then it is a topological embedding, and
- if f is a bijectionBijectionA bijection is a function giving an exact pairing of the elements of two sets. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements...
, then it is a homeomorphismHomeomorphismIn the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...
.
In the first two cases, being open or closed is merely a sufficient condition for the result to follow. In the third case it is necessary as well.
Open and closed mapping theorems
It is useful to have conditions for determining when a map is open or closed. The following are some results along these lines.The closed map lemma states that every continuous function f : X → Y from a compact space
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
X to a Hausdorff space
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
Y is closed and proper
Proper map
In mathematics, a continuous function between topological spaces is called proper if inverse images of compact subsets are compact. In algebraic geometry, the analogous concept is called a proper morphism.- Definition :...
(i.e. preimages of compact sets are compact). A variant of this result states that if a continuous function between locally compact
Locally compact space
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.-Formal definition:...
Hausdorff spaces is proper, then it is also closed.
In functional analysis
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 and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
, the open mapping theorem
Open mapping theorem (functional analysis)
In functional analysis, the open mapping theorem, also known as the Banach–Schauder theorem , is a fundamental result which states that if a continuous linear operator between Banach spaces is surjective then it is an open map...
states that every surjective continuous linear operator between Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
s is an open map.
In complex analysis
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 useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
, the identically named open mapping theorem
Open mapping theorem (complex analysis)
In complex analysis, the open mapping theorem states that if U is a connected open subset of the complex plane C and f : U → C is a non-constant holomorphic function, then f is an open map .The open mapping theorem points to the sharp difference between holomorphy and real-differentiability...
states that every non-constant holomorphic function
Holomorphic function
In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...
defined on a connected
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
open subset of the complex plane
Complex plane
In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...
is an open map.
The invariance of domain
Invariance of domain
Invariance of domain is a theorem in topology about homeomorphic subsets of Euclidean space Rn. It states:The theorem and its proof are due to L.E.J. Brouwer, published in 1912...
theorem states that a continuous and locally injective function between two n-dimensional topological manifolds
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
must be open.