# Quotient space

{{#invoke:Hatnote|hatnote}} Illustration of quotient space, S2, obtained by gluing the boundary (in blue) of the disk D2 to a single point.

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

Let (X,τX) be a topological space, and let ~ be an equivalence relation on X. The quotient space, $Y=X/\!\!\sim$ is defined to be the set of equivalence classes of elements of X:

$Y=\{[x]:x\in X\}=\{\{v\in X:v\sim x\}:x\in X\},$ equipped with the topology where the open sets are defined to be those sets of equivalence classes whose unions are open sets in X:

$\tau _{Y}=\{U\subseteq Y:\bigcup U\in \tau _{X}\}.$ Equivalently, we can define them to be those sets with an open preimage under the quotient map $q:X\to X/\!\!\sim$ which sends a point in X to the equivalence class containing it.

$\tau _{Y}=\{U\subseteq Y:q^{-1}(U)\in \tau _{X}\}.$ The quotient topology is the final topology on the quotient space with respect to the quotient map.

## 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. Such gluing is commonly referred to as the wedge sum.
• 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 fq 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 = fq. 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 : 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 that 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