Quotient space (topology)

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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 ...
and related areas of
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 ...
, the quotient space of a
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
under a given
equivalence relation 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 i ...
is a new topological space constructed by endowing the
quotient set In mathematics, when the elements of some Set (mathematics), 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 ...
of the original topological space with the quotient topology, that is, with the finest topology that makes continuous the canonical projection map (the function that maps points to their
equivalence class 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 ...
es). In other words, a subset of a quotient space is
open 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 Australia ...
if and only if its
preimage In mathematics, the image of a Function (mathematics), function is the set of all output values it may produce. More generally, evaluating a given function f at each Element (mathematics), element of a given subset A of its Domain of a functio ...
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 Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
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 (geometry), chord of the circle. Both definitions a ...
produces the
projective plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane (geometry), plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, paral ...
as a quotient space.

Definition

Let $\left\left(X, \tau_X\right\right)$ be a
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
, and let $\,\sim\,$ be an
equivalence relation 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 i ...
on $X.$ The
quotient set In mathematics, when the elements of some Set (mathematics), 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 ...
, $Y = X / \sim\,$ is the set of
equivalence class 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 ...
es of elements of $X.$ The equivalence class of $x \in X$ is denoted The , , associated with $\,\sim\,$ refers to the following
surjective 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 m ...
map: 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 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 ...
s are the
subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
s $U \subseteq Y = X /$ such that $\ = \cup_ u$ is an
open subset In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a Set (mathematics), set along with a metric (mathematics), distance defined between any two points), open sets are the sets that, with every ...
of $\left\left(X, \tau_X\right\right);$ 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 In mathematics, the image of a Function (mathematics), function is the set of all output values it may produce. More generally, evaluating a given function f at each Element (mathematics), element of a given subset A of its Domain of a functio ...
under the canonical map $q : X \to X /$ (which is defined by

Quotient map

A map $f : X \to Y$ is a quotient map (sometimes called an identification map) if it is
surjective 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 m ...
, and a subset $V \subseteq Y$ is open if and only if $f^\left(V\right)$ 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 $f^\left(D\right)$ 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 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 In chemistry, a saturated compound is a chemical co ...
(with respect to $f$) if it is of the form $S = f^\left(T\right)$ for some set $T,$ which is true if and only if $f^\left(f\left(S\right)\right) = S.$ The assignment $T \mapsto f^\left(T\right)$ establishes a
one-to-one correspondence 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 ...
(whose inverse is $S \mapsto f\left(S\right)$) between subsets $T$ of $Y = f\left(X\right)$ 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 $f\left(S\right)$ 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 $X,$ $f\left(S\right)$ being open in $f\left(X\right)$ implies $S$ is open in $X$). Indeed, if $\tau$ is a
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 ...
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 $f : \left(X, \tau\right) \to Y$ is a quotient map (respectively, continuous) if and only if the same is true of $f : \left\left(X, \tau_f\right\right) \to Y.$ 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 $f\left(S\right)$ 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 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 i ...
$\,\sim\,$ on $X,$ denote the
equivalence class 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 ...
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 class 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 ...
es (that is, it is defined by
surjective map In mathematics, a surjective function (also known as surjection, or onto function) is a Function (mathematics), function that every element can be mapped from element so that . In other words, every element of the function's codomain is the Im ...
and for all $a, b \in X,$ $a \,\sim\, b$ if and only if $q\left(a\right) = q\left(b\right);$ consequently, $q\left(x\right) = q^\left(q\left(x\right)\right)$ 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 Two Mathematical object, mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes of ''a'' and ''b'' wi ...
a
homeomorphism In the mathematics, mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and Continuous function#Continuous functions between topological spaces, continuous function between topologic ...
, 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 $f\left(a\right) = f\left(b\right).$ Then $\,\sim\,$ is an equivalence relation on $X$ such that for every $x \in X,$ which implies that (defined by ) is a
singleton set 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 ...
; denote the unique element in by (so by definition, ). The assignment defines a
bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function (mathematics), function between the elements of two set (mathematics), sets, where each element of one set is pair ...
$\hat : X / \;\;\to\; Y$ between the fibers of $f$ and points in $Y.$ Define the map $q : X \to X /$ as above (by
homeomorphism In the mathematics, mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and Continuous function#Continuous functions between topological spaces, continuous function between topologic ...
(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 $f\big\vert_ ~:~ f^\left(T\right) \to T$ 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
homeomorphic In the mathematics, mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and Continuous function#Continuous functions between topological spaces, continuous function between topologic ...
to the
sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
$S^2.$ * Adjunction space. 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 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 measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
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 bicondi ...
$x - y$ 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 language of ...
. Then the quotient space $X / \sim$ is
homeomorphic In the mathematics, mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and Continuous function#Continuous functions between topological spaces, continuous function between topologic ...
to the
unit circle 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 i ...
$S^1$ via the homeomorphism which sends the equivalence class of $x$ to $\exp\left(2 \pi i x\right).$ * A generalization of the previous example is the following: Suppose a
topological group 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 ...
$G$
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 The gospel, ...
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 physical body, object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an satellite, artificial satellite around an object or pos ...
. 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 (mathematics), division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to 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 with the set $\ \cup \left(\R \setminus \Z\right)$) 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, mobile ...
(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 (mathematics), function whose domain is the natu ...
) in $X.$ Let
homeomorphism In the mathematics, mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and Continuous function#Continuous functions between topological spaces, continuous function between topologic ...
. Let $I = \N$ and let $a_ := \left\left(a_i\right\right)_ \text b_ := \left\left(b_i\right\right)_$ be any sequences (or more generally, any nets) valued in $\left(0, 1\right)$ such that $a_ \to 0 \text b_ \to 1$ in 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
directed set In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty Set (mathematics), set A together with a Reflexive relation, reflexive and Transitive relation, transitive binary relation \,\leq\, (that is, a preorder), with ...
, and making $I := A \times \$ into a net by declaring that for any $\left(a, m\right), \left(b, n\right) \in I,$ $\left(m, a\right) \; \leq \; \left(n, b\right)$ 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

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 fro ...
: if $g : X \to Z$ is a continuous map such that $a \sim b$ implies $g\left(a\right) = g\left(b\right)$ 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) 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 Australia ...
or closed. Note that these conditions are only sufficient, 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 neighbourhood (mathematics), neighborhood not containing the other point. An R0 space is one in which thi ...
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) 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 Australia ...
, 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 neighbourhood (mathematics), neighbourhoods of each which are disjoint s ...
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-seemin ...
$X \times X.$ Connectedness * If a space is connected or path connected, then so are all its quotient spaces. * A quotient space of a
simply connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every Path (topology), path between two points can be continuously transformed (intuitively for embedded spaces, staying ...
or
contractible In mathematics, a topological space ''X'' is contractible if the identity function, 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 ...
space need not share those properties.
Compactness In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed set, closed and bounded set, bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or ...
* If a space is compact, then so are all its quotient spaces. * A quotient space of a
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 ...
space need not be locally compact.
Dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
* 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 topological invariant, topologically invariant way. Informal discussion F ...
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 of a function, range contains the entire 2-dimensional unit square (or more generally an ''n''-dimensional unit hypercube). Because Giuseppe Peano (1858–1932) was the first t ...
s provide such examples.