# Quotient space (topology)

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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**, is defined to be the set of equivalence classes of elements of *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*:

Equivalently, we can define them to be those sets with an open preimage under the surjective map which sends a point in *X* to the equivalence class containing it:

The quotient topology is the final topology on the quotient space with respect to the map .

## Quotient map

A map is **a quotient map** if it is surjective, continuous, and a subset *U* of *Y* is open if and only if is open. Equivalently, is a quotient map if it is onto and is equipped with the final topology with respect to .

Given an equivalence relation on , the canonical map is a quotient map.

## Examples

**Gluing.**Often, topologists talk of gluing points together. If*X*is a topological space and points 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
*I*^{2}= [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*I*^{2}/~ is homeomorphic to the unit sphere*S*^{2}. **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: .- Consider the set 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*S*^{1}via the homeomorphism which sends the equivalence class of*x*to exp(2π*ix*). - 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*S*^{1}.

*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* : *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*.

- In general, quotient spaces are ill-behaved with respect to separation axioms. The separation properties of
- 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

- Topological space
- Subspace (topology)
- Product space
- Disjoint union (topology)
- Final topology
- Mapping cone

### Algebra

## References

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