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.

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.

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 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 (sometimes called an identification map[1]) 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 (closed) if and only if is open (resp. closed). Every quotient map is continuous but not every continuous map is a quotient map.

Saturated sets

A subset of is called saturated (with respect to ) if it is of the form for some set which is true if and only if The assignment establishes a one-to-one correspondence (whose inverse is ) between subsets of and saturated subsets of With this terminology, a surjection is a quotient map if and only if for every saturated subset of is open in if and only if is open in In particular, open subsets of that are not saturated have no impact on whether the function is a quotient map (or, indeed, continuous: a function is continuous if and only if, for every saturated such that is open in , the set is open in ).

Indeed, if is a topology on and is any map, then the set of all that are saturated subsets of forms a topology on If is also a topological space then is a quotient map (respectively, continuous) 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 (that is, it is defined by for every ) is called the canonical map. 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 (not yet assumed to be continuous or a quotient map) and declare for all that if and only if Then is an equivalence relation on such that for every which implies that (defined by ) is a singleton set; denote the unique element in by (so by definition, ). The assignment defines a bijection between the fibers of and points in Define the map as above (by ) and give the quotient topology induced by (which makes a quotient map). 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 (or equivalently, if and only if both and its inverse are continuous).

Related definitions

A hereditarily quotient map 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

For example, is homeomorphic to the circle

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.

Characteristic property of the quotient topology
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 (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 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

Separation

Connectedness

Compactness

Dimension

See also

Topology

Algebra

Notes

  1. ^ Brown 2006, p. 103.

References