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.
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.
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
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
A map is a quotient map (sometimes called an identification map) 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 (although always holds for every subset equality is in general not guaranteed; and a non-saturated set exists if and only if is not injective).
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").
In fact, 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 so that the image of this set is is a singleton set; let denote its unique element (so by definition, ).
This 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).
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.
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 (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
For example, is homeomorphic to the circle
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:
A generalization of the previous example is the following: Suppose a topological groupacts 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 (that is identified as a single point) then the quotient (which is identifiable with the set ) is a countably infinite bouquet of circles joined at a single point
This next example shows that it is in general not true that if is a quotient map then every convergent sequence (respectively, every convergentnet) in has a lift (by ) to a convergent sequence (or convergent net) in Let and Let and let be the quotient map so that and for every The map defined by is well-defined (because ) and a homeomorphism. Let and let be any sequences (or more generally, any nets) 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 (that is, there is no sequence in that both converges to some and satisfies for every ). 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 (1) and (2) if then the -indexed net defined by letting equal and equal to has no lift (by ) to a convergent -indexed net in
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.
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.