Quotient space (topology)

In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient topology, that is, with the finest topology that makes continuous the canonical projection map (the function that maps points to their equivalence classes). In other words, a subset of a quotient space is open if and only if its preimage under the canonical projection map is open in the original topological space.

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

Illustration of the construction of a topological sphere as the quotient space of a disk, by gluing together to a single point the points (in blue) of the boundary of the disk.

Intuitively speaking, the points of each equivalence class are identified or "glued together" for forming a new topological space. For example, identifying the points of a sphere that belong to the same diameter produces the projective plane as a quotient space.

Definition

Let (X, τ_X) be a topological space, and let ~ be an equivalence relation on X. The quotient set, Y = X / ~ is the set of equivalence classes of elements of X. As usual, the equivalence class of x ∈ X is denoted [x].

The quotient space under ~ is the quotient set Y equipped with the quotient topology, that is the topology whose open sets are the subsets U ⊆ Y such that ${\displaystyle \{x\in X:[x]\in U\}}$ is open in X. That is,

${\displaystyle \tau _{Y}=\left\{U\subseteq Y:\{x\in X:[x]\in U\}\in \tau _{X}\right\}.}$

Equivalently, the open sets of the quotient topology are the subsets of Y that have an open preimage under the surjective map x → [x].

The quotient topology is the final topology on the quotient set, with respect to the map x → [x].

Quotient map

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

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

Examples

• Gluing. Topologists talk of gluing points together. If X is a topological space, gluing the points x and y in X means considering 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 sphere S².

For example, ${\displaystyle [0,1]/\{0,1\}}$ is homeomorphic to the circle ${\displaystyle S^{1}}$.

• 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 a closed disc with its boundary identified to a single point: ${\displaystyle D^{2}/\partial {D^{2}}}$.
• Consider the set R of 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 via addition, then the quotient is the circle. However, if Z is thought of as a subspace of R, then the quotient is a countably 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.

Characteristic property of the quotient topology

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 copiously 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. For topological groups, the quotient map is open.

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.

See also

Topology

• Disjoint union (topology)
• Final topology – finest topology making some functions continuous
• Mapping cone (topology)
• Product space
• Subspace (topology)
• Topological space – Mathematical structure

Algebra

• Mapping cone (homological algebra) – Tool in homological algebra
• Quotient category
• Quotient group
• Quotient space (linear algebra)

References

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