# Quotient space (topology)

{{#invoke:Hatnote|hatnote}} Illustration of quotient space, S2, obtained by gluing the boundary (in blue) of the disk D2 together 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 topological 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. The quotient topology consists of all sets with an open preimage under the canonical projection map that maps each element to its equivalence class.

## 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:

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

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

## Examples

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 q : XY 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 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