Home > Quotient space
:For quotient spaces in linear algebra, see quotient space (linear algebra).In topology and related areas of mathematics, 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 relation. This is commonly done in order to construct new spaces from given ones.
1 Definition
Suppose X is a topological space and ~ is an equivalence relation on X. We define a topology on the quotient set X/~ (the set consisting of all equivalence classes of ~) as follows: a set of equivalence classes in X/~ is open if and only if their union is open in X.
This is the quotient topology on the quotient set X/~.
Equivalently, the quotient topology can be characterized in the following manner: Let q : X → X/~ be the projection map which sends each element of X to its equivalence class. Then the quotient topology on X/~ is the finest topology for which q is continuous.
Given a surjective map f : X → Y from a topological space X to a set Y we can define the quotient topology on Y as the finest topology for which f is continuous. This is equivalent to saying that a subset V ⊆ Y is open in Y if and only if its preimage f−1(V) is open in X. The map f induces an equivalence relation on X by saying x~y iff f(x) = f(y). The quotient space X/~ is then homeomorphic to Y (with its quotient topology) via the homeomorphism which sends the equivalence class of x to f(x).
In general, a surjective, continuous map f : X → Y is said to be a quotient map if Y has the quotient topology determined by f.
2 Examples
- 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 sphereFor other uses, see sphere (disambiguation). A sphere is, roughly speaking, a ball-shaped object. In non-mathematical usage a sphere is often considered to be solid (which mathematicians call ball . But in mathematics, a sphere is the boundary of a ball, S2.
- 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 indentified to a single point: D2/∂D2.
- Consider the set X = R of all real numberIn mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. The term "real number" is a retronym coined in response to " imaginary number". Real numbers mays with the ordinary topology, and write x ~ y iff x−y is an integerThe integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (−1, −2, −3,. and the number zero. The set of all integers is usually denoted in mathematics by Z (or Z in blackboard bold, ), which st. Then the quotient space X/~ is homeomorphic to the unit circleIllustration of a unit circle. t is an angle measure. In mathematics, a unit circle is a circle with unit radius, i. a circle whose radius is 1. Frequently, especially in trigonometry, "the" unit circle is the circle of radius 1 centered at the origin (0, 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 G acts continuously on a space X. One can form an equivalence relation on X by saying points are equivalent iff 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 circles joined at a single point.
