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.

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.

is the set of equivalence classes of elements of

equipped with the quotient topology, whose open sets are those subsets

The quotient topology is the final topology on the quotient set, with respect to the map

is equipped with the final topology induced by

The latter condition admits two more-elementary formulations: a subset

establishes a one-to-one correspondence (whose inverse is

is a quotient map if and only if for every saturated subset

is a quotient map (or, indeed, continuous: a function

is a quotient map (respectively, continuous) if and only if the same is true of

Quotient space of fibers characterization Given an equivalence relation

denote the equivalence class of a point

denote the set of equivalence classes.

that sends points to their equivalence classes (that is, it is defined by

In particular, this shows that the set of equivalence class

is exactly the set of fibers of the canonical map

the quotient topology induced by

Up to a homeomorphism, this construction is representative of all quotient spaces; the precise meaning of this is now explained.

be a surjection between topological spaces (not yet assumed to be continuous or a quotient map) and declare for all

) is a singleton set; denote the unique element in

are characterized among surjective maps by the following property: if

then there exists a unique continuous map

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.

It is easy to construct examples of quotient maps that are neither open nor closed.

For topological groups, the quotient map is open.

Separation Connectedness Compactness Dimension Topology Algebra