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.
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.


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 quotient, canonical, projection map associated with refers to the following surjective map:

For any subset (so in particular, for every ) the following holds:

The quotient space under is the quotient set equipped with the quotient topology, which is the topology whose open sets are the subsets such that is an open subset of that is, is open in the quotient topology on if and only if Thus,

Equivalently, the open sets of the quotient topology are the subsets of that have an open preimage under the canonical map (which is defined by ). Similarly, a subset is closed in if and only if is a closed subset of

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 a subset is open if and only if is open. Equivalently, a surjection is a quotient map if and only if for every subset is closed in if and only if is closed in

Final topology definition

Alternatively, is a quotient map if it is onto and is equipped with the final topology with respect to

Saturated sets and quotient maps

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 or not the function is a quotient map; non-saturated subsets are irrelevant to the definition of "quotient map" just as they are irrelevant to the open-set definition of continuity (because a function is continuous if and only if for every saturated subset of being open in implies is open in ). Indeed, if is a topology on and is any map then 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

Every quotient map is continuous but not every continuous map is a quotient map. A continuous surjection fails to be a quotient map if and only if has some saturated open subset such that is not open in (this statement remains true if both instances of the word "open" are replaced with "closed").

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.


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


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

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:

Universal Property of Quotient Spaces.svg

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





See also




  1. ^ Brown 2006, p. 103.