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topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
and related areas of mathematics, the quotient space of a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
under a given equivalence relation is a new topological space constructed by endowing the
quotient set In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
of the original topological space with the quotient topology, that is, with the finest topology that makes
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 ...
the
canonical projection map In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...
(the function that maps points to their equivalence classes). In other words, a subset of a quotient space is
open Open or OPEN may refer to: Music * Open (band), Australian pop/rock band * The Open (band), English indie rock band * ''Open'' (Blues Image album), 1969 * ''Open'' (Gotthard album), 1999 * ''Open'' (Cowboy Junkies album), 2001 * ''Open'' ( ...
if and only if its preimage under the canonical projection map is open in the original topological space. Intuitively speaking, the points of each equivalence class are or "glued together" for forming a new topological space. For example, identifying the points of a
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ...
that belong to the same
diameter In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid fo ...
produces the
projective plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that d ...
as a quotient space.


Definition

Let \left(X, \tau_X\right) be a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
, and let \,\sim\, be an equivalence relation on X. The
quotient set In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
, Y = X / \sim\, is the set of equivalence classes of elements of X. The equivalence class of x \in X is denoted The , , associated with \,\sim\, refers to the following surjective map: \begin q :\;&& X &&~\to & ~X/ \\ .3ex && x &&~\mapsto& ~ \end For any subset S \subseteq X / (so in particular, s \subseteq X for every s \in S) the following holds: q^(S)=\ = \bigcup_ s. The quotient space under \,\sim\, is the quotient set Y equipped with the quotient topology, which is the topology whose
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
s are the subsets U \subseteq Y = X / such that \ = \cup_ u is an open subset of \left(X, \tau_X\right); that is, U \subseteq X / is open in the quotient topology on X / if and only if \ \in \tau_X. Thus, \tau_Y = \left\. Equivalently, the open sets of the quotient topology are the subsets of Y that have an open preimage under the canonical map q : X \to X / (which is defined by q(x) = /math>). Similarly, a subset S \subseteq X / is closed in X / if and only if \ is a closed subset of \left(X, \tau_X\right). The quotient topology is the
final topology In general topology and related areas of mathematics, the final topology (or coinduced, strong, colimit, or inductive topology) on a set X, with respect to a family of functions from topological spaces into X, is the finest topology on X that make ...
on the quotient set, with respect to the map x \mapsto


Quotient map

A map f : X \to Y is a quotient map (sometimes called an identification map) if it is surjective, and a subset V \subseteq Y is open if and only if f^(V) is open. Equivalently, a surjection f : X \to Y is a quotient map if and only if for every subset D \subseteq Y, D is closed in Y if and only if f^(D) is closed in X. Final topology definition Alternatively, f is a quotient map if it is onto and Y is equipped with the
final topology In general topology and related areas of mathematics, the final topology (or coinduced, strong, colimit, or inductive topology) on a set X, with respect to a family of functions from topological spaces into X, is the finest topology on X that make ...
with respect to f. Saturated sets and quotient maps A subset S of X is called
saturated Saturation, saturated, unsaturation or unsaturated may refer to: Chemistry * Saturation, a property of organic compounds referring to carbon-carbon bonds ** Saturated and unsaturated compounds **Degree of unsaturation ** Saturated fat or fatty ac ...
(with respect to f) if it is of the form S = f^(T) for some set T, which is true if and only if f^(f(S)) = S. The assignment T \mapsto f^(T) establishes a
one-to-one correspondence 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 ...
(whose inverse is S \mapsto f(S)) between subsets T of Y = f(X) and saturated subsets of X. With this terminology, a surjection f : X \to Y is a quotient map if and only if for every subset S of X, S is open in X if and only if f(S) is open in Y. In particular, open subsets of X that are saturated have no impact on whether or not the function f is a quotient map; non-saturated subsets are irrelevant to the definition of "quotient map" just as they are irrelevant to the open-set definition of continuity (because a function f : X \to Y is continuous if and only if for every subset S of X, f(S) being open in f(X) implies S is open in X). Indeed, if \tau is a
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
on X and f : X \to Y is any map then set \tau_f of all U \in \tau that are saturated subsets of X forms a topology on X. If Y is also a topological space then f : (X, \tau) \to Y is a quotient map (respectively,
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 ...
) if and only if the same is true of f : \left(X, \tau_f\right) \to Y. Every quotient map is continuous but not every continuous map is a quotient map. A continuous surjection f : X \to Y to be a quotient map if and only if X has some open subset S such that f(S) is open in Y (this statement remains true if both instances of the word "open" are replaced with "closed"). Quotient space of fibers characterization Given an equivalence relation \,\sim\, on X, denote the equivalence class of a point x \in X by := \ and let X / := \ denote the set of equivalence classes. The map q : X \to X / that sends points to their equivalence classes (that is, it is defined by q(x) := /math> for every x \in X) is called . It is a
surjective map 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 ...
and for all a, b \in X, a \,\sim\, b if and only if q(a) = q(b); consequently, q(x) = q^(q(x)) for all x \in X. In particular, this shows that the set of equivalence class X / is exactly the set of fibers of the canonical map q. If X is a topological space then giving X / the quotient topology induced by q will make it into a quotient space and make q : X \to X / into a quotient map. Up to a
homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...
, this construction is representative of all quotient spaces; the precise meaning of this is now explained. Let f : X \to Y be a surjection between topological spaces (not yet assumed to be continuous or a quotient map) and declare for all a, b \in X that a \,\sim\, b if and only if f(a) = f(b). Then \,\sim\, is an equivalence relation on X such that for every x \in X, = f^(f(x)), which implies that f( (defined by f( = \) is a
singleton set In mathematics, a singleton, also known as a unit set or one-point set, is a set with exactly one element. For example, the set \ is a singleton whose single element is 0. Properties Within the framework of Zermelo–Fraenkel set theory, the ...
; denote the unique element in f( by \hat( (so by definition, f( = \). The assignment \mapsto \hat( defines a bijection \hat : X / \;\;\to\; Y between the fibers of f and points in Y. Define the map q : X \to X / as above (by q(x) := /math>) and give X / \sim the quotient topology induced by q (which makes q a quotient map). These maps are related by: f = \hat \circ q \quad \text \quad q = \hat^ \circ f. From this and the fact that q : X \to X / \sim is a quotient map, it follows that f : X \to Y is continuous if and only if this is true of \hat : X / \sim \;\;\to\; Y. Furthermore, f : X \to Y is a quotient map if and only if \hat : X / \sim \;\;\to\; Y is a
homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...
(or equivalently, if and only if both \hat and its inverse are continuous).


Related definitions

A is a surjective map f : X \to Y with the property that for every subset T \subseteq Y, the restriction f\big\vert_ ~:~ f^(T) \to T is also a quotient map. There exist quotient maps that are not hereditarily quotient.


Examples

* Gluing. Topologists talk of gluing points together. If X is a topological space, gluing the points x and y in X means considering the quotient space obtained from the equivalence relation a \sim b if and only if a = b or a = x, b = y (or a = y, b = x). * Consider the unit square I^2 =
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
\times
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
/math> and the equivalence relation ~ generated by the requirement that all boundary points be equivalent, thus identifying all boundary points to a single equivalence class. Then I^2 / \sim is homeomorphic to the
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ...
S^2. *
Adjunction space In mathematics, an adjunction space (or attaching space) is a common construction in topology where one topological space is attached or "glued" onto another. Specifically, let ''X'' and ''Y'' be topological spaces, and let ''A'' be a subspace of ' ...
. More generally, suppose X is a space and A is a subspace of X. One can identify all points in A to a single equivalence class and leave points outside of A equivalent only to themselves. The resulting quotient space is denoted X/A. The 2-sphere is then homeomorphic to a
closed disc In geometry, a disk (also spelled disc). is the region in a plane bounded by a circle. A disk is said to be ''closed'' if it contains the circle that constitutes its boundary, and ''open'' if it does not. For a radius, r, an open disk is usu ...
with its boundary identified to a single point: D^2 / \partial. * Consider the set \R of
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s with the ordinary topology, and write x \sim y
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is b ...
x - y is an
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
. Then the quotient space X / \sim is homeomorphic to 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 ...
S^1 via the homeomorphism which sends the equivalence class of x to \exp(2 \pi i x). * A generalization of the previous example is the following: Suppose a
topological group In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two st ...
G
acts The Acts of the Apostles ( grc-koi, Πράξεις Ἀποστόλων, ''Práxeis Apostólōn''; la, Actūs Apostolōrum) is the fifth book of the New Testament; it tells of the founding of the Christian Church and the spread of its message ...
continuously on a space X. One can form an equivalence relation on X by saying points are equivalent if and only if they lie in the same
orbit In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as ...
. The quotient space under this relation is called the orbit space, denoted X / G. In the previous example G = \Z acts on \R by translation. The orbit space \R / \Z is homeomorphic to S^1. **''Note'': The notation \R / \Z is somewhat ambiguous. If \Z is understood to be a group acting on \R via addition, then the
quotient In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
is the circle. However, if \Z is thought of as a topological subspace of \R (that is identified as a single point) then the quotient \ \cup \ (which is
identifiable In statistics, identifiability is a property which a model must satisfy for precise inference to be possible. A model is identifiable if it is theoretically possible to learn the true values of this model's underlying parameters after obtaining an ...
with the set \ \cup (\R \setminus \Z)) is a countably infinite bouquet of circles joined at a single point \Z. * This next example shows that it is in general true that if q : X \to Y is a quotient map then every
convergent sequence As the positive integer n becomes larger and larger, the value n\cdot \sin\left(\tfrac1\right) becomes arbitrarily close to 1. We say that "the limit of the sequence n\cdot \sin\left(\tfrac1\right) equals 1." In mathematics, the limit ...
(respectively, every convergent
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 Y has a
lift Lift or LIFT may refer to: Physical devices * Elevator, or lift, a device used for raising and lowering people or goods ** Paternoster lift, a type of lift using a continuous chain of cars which do not stop ** Patient lift, or Hoyer lift, mobil ...
(by q) to a convergent sequence (or
convergent net In mathematics, more specifically in general topology and related branches, a net or Moore–Smith sequence is a generalization of the notion of a sequence. In essence, a sequence is a function whose domain is the natural numbers. The codoma ...
) in X. Let X =
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
/math> and \,\sim ~=~ \ ~\cup~ \left\. Let Y := X /\,\sim\, and let q : X \to X / \sim\, be the quotient map q(x) := so that q(0) = q(1) = \ and q(x) = \ for every x \in (0, 1). The map h : X / \,\sim\, \to S^1 \subseteq \Complex defined by h( := e^ is well-defined (because e^ = 1 = e^) and a
homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...
. Let I = \N and let a_ := \left(a_i\right)_ \text b_ := \left(b_i\right)_ be any sequences (or more generally, any nets) valued in (0, 1) such that a_ \to 0 \text b_ \to 1 in X =
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
Then the sequence y_1 := q\left(a_1\right), y_2 := q\left(b_1\right), y_3 := q\left(a_2\right), y_4 := q\left(b_2\right), \ldots converges to = /math> in X / \sim\, but there does not exist any convergent lift of this sequence by the quotient map q (that is, there is no sequence s_ = \left(s_i\right)_ in X that both converges to some x \in X and satisfies y_i = q\left(s_i\right) for every i \in I). This counterexample can be generalized to nets by letting (A, \leq) be any
directed set In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive and transitive binary relation \,\leq\, (that is, a preorder), with the additional property that every pair of elements ha ...
, and making I := A \times \ into a net by declaring that for any (a, m), (b, n) \in I, (m, a) \; \leq \; (n, b) holds if and only if both (1) a \leq b, and (2) if a = b \text m \leq n; then the A-indexed net defined by letting y_ equal a_i \text m = 1 and equal to b_i \text m = 2 has no lift (by q) to a convergent A-indexed net in X =
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...


Properties

Quotient maps q : X \to Y are characterized among surjective maps by the following property: if Z is any topological space and f : Y \to Z is any function, then f is continuous if and only if f \circ q is continuous. The quotient space X / together with the quotient map q : X \to X / is characterized by the following
universal property In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fr ...
: if g : X \to Z is a continuous map such that a \sim b implies g(a) = g(b) for all a, b \in X, then there exists a unique continuous map f : X / \to Z such that g = f \circ q. In other words, the following diagram commutes: One says that g ''descends to the quotient'' for expressing this, that is that it factorizes through the quotient space. The continuous maps defined on X / are, therefore, precisely those maps which arise from continuous maps defined on X that respect the equivalence relation (in the sense that they send equivalent elements to the same image). This criterion is copiously used when studying quotient spaces. Given a continuous surjection q : X \to Y it is useful to have criteria by which one can determine if q is a quotient map. Two sufficient criteria are that q be
open Open or OPEN may refer to: Music * Open (band), Australian pop/rock band * The Open (band), English indie rock band * ''Open'' (Blues Image album), 1969 * ''Open'' (Gotthard album), 1999 * ''Open'' (Cowboy Junkies album), 2001 * ''Open'' ( ...
or closed. Note that these conditions are only
sufficient 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 ...
, not necessary. It is easy to construct examples of quotient maps that are neither open nor closed. For topological groups, the quotient map is open.


Compatibility with other topological notions

Separation * In general, quotient spaces are ill-behaved with respect to separation axioms. The separation properties of X need not be inherited by X / \sim, and X / \sim may have separation properties not shared by X. * X / \sim is a
T1 space In topology and related branches of mathematics, a T1 space is a topological space in which, for every pair of distinct points, each has a neighborhood not containing the other point. An R0 space is one in which this holds for every pair of to ...
if and only if every equivalence class of \,\sim\, is closed in X. * If the quotient map is
open Open or OPEN may refer to: Music * Open (band), Australian pop/rock band * The Open (band), English indie rock band * ''Open'' (Blues Image album), 1969 * ''Open'' (Gotthard album), 1999 * ''Open'' (Cowboy Junkies album), 2001 * ''Open'' ( ...
, then X / \sim is a
Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the m ...
if and only if ~ is a closed subset of the
product space 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. This topology differs from another, perhaps more natural-seemi ...
X \times X.
Connectedness In mathematics, connectedness is used to refer to various properties meaning, in some sense, "all one piece". When a mathematical object has such a property, we say it is connected; otherwise it is disconnected. When a disconnected object can be s ...
* If a space is connected or path connected, then so are all its quotient spaces. * A quotient space of a simply connected or
contractible In mathematics, a topological space ''X'' is contractible if the identity map on ''X'' is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within th ...
space need not share those properties. Compactness * If a space is compact, then so are all its quotient spaces. * A quotient space of a locally compact space need not be locally compact.
Dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
* The
topological dimension In mathematics, the Lebesgue covering dimension or topological dimension of a topological space is one of several different ways of defining the dimension of the space in a topologically invariant way. Informal discussion For ordinary Euclidean ...
of a quotient space can be more (as well as less) than the dimension of the original space;
space-filling curve In mathematical analysis, a space-filling curve is a curve whose range contains the entire 2-dimensional unit square (or more generally an ''n''-dimensional unit hypercube). Because Giuseppe Peano (1858–1932) was the first to discover one, spa ...
s provide such examples.


See also

Topology * * * * * * * Algebra * * * *


References

* * * * * * * * {{reflist, group=proof Theory of continuous functions General topology Group actions (mathematics) Space (topology) Topology