Definitions

# Quotient space

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.

## 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 : XX/~ 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 : XY 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 VY 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 x1~x2 if and only if f(x1) = f(x2). 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 : XY is said to be a quotient map if Y has the quotient topology determined by f.

## Examples

• Gluing. Often, topologists talk of gluing points together. If X is a topological space and points $x,y in X$ 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 S2.
• 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 the unit disc with its boundary identified to a single point: D2/∂D2.
• Consider the set X = R of all real numbers with the ordinary topology, and write x ~ y if and only if xy is an integer. Then the quotient space X/~ is homeomorphic to the unit circle 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 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 circles joined at a single point.

## Properties

Quotient maps q : XY are characterized among surjective maps by the following property: if Z is any topological space and f : YZ 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 : XX/~ is characterized by the following universal property: if g : XZ 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 being used when studying quotient spaces.

Given a continuous surjection f : XY 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 if and only if every equivalence class of ~ is closed in X.
• If the quotient map is open then X/~ is a Hausdorff space if and only if ~ is a closed subset of the product space X×X.
• Connectedness
• 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