Quotient space (topology)

For quotient spaces in linear algebra, see quotient space (linear algebra).

Illustration of quotient space,

S 2

, obtained by gluing the boundary (in blue) of the disk

D 2

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 $(X, τ X)$ 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 $q$ .

Quotient map

A map $f: X\to Y$ is a quotient map if it is surjective, and a subset U of Y is open if and only if $f^{-1}(U)$ is open. Equivalently, $f$ is a quotient map if it is onto and $Y$ is equipped with the final topology with respect to $f$ .

Given an equivalence relation $\sim$ on $X$ , the canonical map $q: X \to X/\!\!\sim$ is a quotient map.

Examples

Gluing. 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).
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: $D^{2}/\partial{D^{2}}$ .
Consider the set $X = \mathbb{R}$ of all real numbers with the ordinary topology, and write x ~ y if and only if x − y 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 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.

Note: 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 : 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 ∘ q is continuous.

The quotient space X/~ together with the quotient map q : X → X/~ is characterized by the following universal property: 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 ∘ 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 q : X → Y 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
- 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
- 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
- The topological dimension of a quotient space can be more (as well as less) than the dimension of the original space; space-filling curves provide such examples.

References

Willard, Stephen (1970). General Topology. Reading, MA: Addison-Wesley. ISBN 0-486-43479-6.
Quotient space at PlanetMath.org.