In
topologyTopology 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...
and related areas of
mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a
quotient space (also called an
identification space) is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an
equivalence relationIn mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...
. This is commonly done in order to construct new spaces from given ones.
Definition
Let be a
topological spaceTopological 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...
, and let ~ be an
equivalence relationIn mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...
on
X. The
quotient space,
is defined to be the set of
equivalence classes of elements of :
equipped with the topology where the open sets are defined to be those sets of equivalence classes whose unions are open sets in
X:
Equivalently, we can define them to be those sets with an open preimage under the quotient map
which sends a point in
X to the equivalence class containing it.
Examples
- Gluing. Often, topologists talk of gluing points together. If X is a topological space and points
are to be "glued", then what is meant is that we are to consider the quotient space obtained from the equivalence relation a ~ b if and only if a = b or a = x, b = y (or a = y, b = x). The two points are henceforth interpreted as one point.
- Consider the unit square I2 = [0,1]×[0,1] 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 I2/~ is homeomorphic to the unit sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
S2.
- Adjunction space
In mathematics, an adjunction space is a common construction in topology where one topological space is attached or "glued" onto another. Specifically, let X and Y be a topological spaces with A a subspace of Y. Let f : A → X be a continuous map...
. 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 the unit disc with its boundary identified to a single point: D2/∂D2.
- Consider the set X = R of all 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...
s with the ordinary topology, and write x ~ y if and only ifIn logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
x−y is an integerThe 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...
. Then the quotient space X/~ is homeomorphic to the unit circleIn 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...
S1 via the homeomorphism which sends the equivalence class of x to exp(2πix).
- A vast generalization of the previous example is the following: Suppose a topological group
In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a...
G actsIn algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
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. 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 S1.
Warning: The notation
R/
Z is somewhat ambiguous. If
Z is understood to be a group acting on
R then the quotient is the circle. However, if
Z is thought of as a subspace of
R, then the quotient is an infinite
bouquet of circlesIn mathematics, a rose is a topological space obtained by gluing together a collection of circles along a single point. The circles of the rose are called petals. Roses are important in algebraic topology, where they are closely related to free groups.- Definition :A rose is a wedge sum of circles...
joined at a single point.
Properties
Quotient maps
q :
X →
Y are characterized among surjective maps by the following property: if
Z is any topological space and
f :
Y →
Z is any function, then
f is continuous if and only if
f O q is continuous.
The quotient space
X/~ together with the quotient map
q :
X →
X/~ is characterized by the following
universal propertyIn various branches of mathematics, a useful construction is often viewed as the “most efficient solution” to a certain problem. The definition of a universal property uses the language of category theory to make this notion precise and to study it abstractly.This article gives a general treatment...
: if
g :
X →
Z is a continuous map such that
a~
b implies
g(
a)=
g(
b) for all
a and
b in
X, then there exists a unique continuous map
f :
X/~ →
Z such that
g =
f O q. We say that
g descends to the quotient.
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 constantly used when studying quotient spaces.
Given a continuous surjection
f :
X →
Y it is useful to have criteria by which one can determine if
f is a quotient map. Two sufficient criteria are that
f be open or closed. Note that these conditions are only sufficient, not necessary. It is easy to construct examples of quotient maps which are neither open nor closed.
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/~, and X/~ may have separation properties not shared by X.
- X/~ 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 an open neighborhood not containing the other. An R0 space is one in which this holds for every pair of topologically distinguishable points...
if and only if every equivalence class of ~ is closed in X.
- If the quotient map is open then X/~ is a 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...
if and only if ~ is a closed subset of the product space X×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...
- If a space is connected or path connected, then so are all its quotient spaces.
- A quotient space of a simply connected or contractible space need not share those properties.
- Compactness
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...
- 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 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 because only one coordinate is needed to specify a point on it...
- The topological dimension 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...
s provide such examples.
Topology
- 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...
- Subspace (topology)
- Product space
- Disjoint union (topology)
In general topology and related areas of mathematics, the disjoint union of a family of topological spaces is a space formed by equipping the disjoint union of the underlying sets with a natural topology called the disjoint union topology...
- Final topology
In general topology and related areas of mathematics, the final topology on a set X, with respect to a family of functions into X, is the finest topology on X which makes those functions continuous.- Definition :Given a set X and a family of topological spaces Y_i with functionsf_i: Y_i \to Xthe...
- Mapping cone
In mathematics, especially homotopy theory, the mapping cone is a construction C_f of topology, analogous to a quotient space. It is also called the homotopy cofiber, and also notated Cf.-Definition:...
Algebra
- Quotient group
In mathematics, specifically group theory, a quotient group is a group obtained by identifying together elements of a larger group using an equivalence relation...
- Quotient space (linear algebra)
In linear algebra, the quotient of a vector space V by a subspace N is a vector space obtained by "collapsing" N to zero. The space obtained is called a quotient space and is denoted V/N ....
- Quotient category
In mathematics, a quotient category is a category obtained from another one by identifying sets of morphisms. The notion is similar to that of a quotient group or quotient space, but in the categorical setting.-Definition:Let C be a category...
- Mapping cone (homological algebra)
In homological algebra, the mapping cone is a construction on a map of chain complexes inspired by the analogous construction in topology. In the theory of triangulated categories it is a kind of combined kernel and cokernel: if the chain complexes take their terms in an abelian category, so that...