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. 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.
Intuitively speaking, the points of each equivalence class are 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 be a topological space, and let be an equivalence relation on The quotient set is the set of equivalence classes of elements of The equivalence class of is denoted
The construction of defines a canonical surjection
As discussed below, is a quotient mapping, commonly called the canonical quotient map, or canonical projection map, associated to
The quotient space under is the set equipped with the quotient topology, whose open sets are those subsets whose preimage is open. In other words, is open in the quotient topology on if and only if is open in Similarly, a subset is closed if and only if is closed in
The quotient topology is the final topology on the quotient set, with respect to the map

Quotient map

A map is a quotient map if it is surjective and is equipped with the final topology induced by The latter condition admits two more-elementary formulations: a subset is open if and only if is open. Every quotient map is continuous but not every continuous map is a quotient map.
Saturated sets
A subset of is called saturated if it is of the form for some set which is true if and only if
The assignment establishes a one-to-one correspondence between subsets of and saturated subsets of
With this terminology, a surjection is a quotient map if and only if for every subset of is open in if and only if is open in
In particular, open subsets of that are saturated have no impact on whether the function is a quotient map if and only if the same is true of
Quotient space of fibers characterization
Given an equivalence relation on denote the equivalence class of a point by and let denote the set of equivalence classes. The map that sends points to their equivalence classes is called. It is a surjective map and for all if and only if consequently, for all In particular, this shows that the set of equivalence class is exactly the set of fibers of the canonical map
If is a topological space then giving the quotient topology induced by will make it into a quotient space and make into a quotient map.
Up to a homeomorphism, this construction is representative of all quotient spaces; the precise meaning of this is now explained.
Let be a surjection between topological spaces and declare for all that if and only if Then is an equivalence relation on such that for every which implies that is a singleton set; denote the unique element in by .
The assignment defines a bijection between the fibers of and points in
Define the map as above and give the quotient topology induced by . These maps are related by:
From this and the fact that is a quotient map, it follows that is continuous if and only if this is true of Furthermore, is a quotient map if and only if is a homeomorphism.

Related definitions

A is a surjective map with the property that for every subset the restriction is also a quotient map.
There exist quotient maps that are not hereditarily quotient.

Examples

Gluing. Topologists talk of gluing points together. If is a topological space, gluing the points and in means considering the quotient space obtained from the equivalence relation if and only if or .
Consider the unit square 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 is homeomorphic to the sphere
Adjunction space. More generally, suppose is a space and is a subspace of One can identify all points in to a single equivalence class and leave points outside of equivalent only to themselves. The resulting quotient space is denoted The 2-sphere is then homeomorphic to a closed disc with its boundary identified to a single point:
Consider the set of real numbers with the ordinary topology, and write if and only if is an integer. Then the quotient space is homeomorphic to the unit circle via the homeomorphism which sends the equivalence class of to
A generalization of the previous example is the following: Suppose a topological group acts continuously on a space One can form an equivalence relation on 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 In the previous example acts on by translation. The orbit space is homeomorphic to
*Note: The notation is somewhat ambiguous. If is understood to be a group acting on via addition, then the quotient is the circle. However, if is thought of as a topological subspace of then the quotient is a countably infinite bouquet of circles joined at a single point
This next example shows that it is in general true that if is a quotient map then every convergent sequence in has a lift to a convergent sequence in Let and Let and let be the quotient map so that and for every The map defined by is well-defined and a homeomorphism. Let and let be any sequences valued in such that in Then the sequence converges to in but there does not exist any convergent lift of this sequence by the quotient map . This counterexample can be generalized to nets by letting be any directed set, and making into a net by declaring that for any holds if and only if both and if then the -indexed net defined by letting equal and equal to has no lift to a convergent -indexed net in
Properties

Quotient maps are characterized among surjective maps by the following property: if is any topological space and is any function, then is continuous if and only if is continuous.
Image:QuotientSpace-01.svg|center|Characteristic property of the quotient topology
The quotient space together with the quotient map is characterized by the following universal property: if is a continuous map such that implies for all then there exists a unique continuous map such that In other words, the following diagram commutes:
One says that descends to the quotient for expressing this, that is that it factorizes through the quotient space. The continuous maps defined on are, therefore, precisely those maps which arise from continuous maps defined on that respect the equivalence relation. This criterion is copiously used when studying quotient spaces.
Given a continuous surjection it is useful to have criteria by which one can determine if is a quotient map. Two sufficient criteria are that 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

In general, quotient spaces are ill-behaved with respect to separation axioms. The separation properties of need not be inherited by and may have separation properties not shared by
is a T1 space if and only if every equivalence class of is closed in
If the quotient map is open, then is a Hausdorff space if and only if ~ is a closed subset of the product space

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 than the dimension of the original space; space-filling curves provide such examples.