Topological space construction
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 quotient, canonical, projection map associated with
refers to the following surjective map:
![{\displaystyle {\begin{alignedat}{4}q:\;&&X&&~\to &~X/{\sim }\\[0.3ex]&&x&&~\mapsto &~[x].\end{alignedat))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1901f37a8dcd98a7708f29fa9b82153c6be89acf)
For any subset
(so in particular,
for every
) the following holds:
![{\displaystyle q^{-1}(S)=\{x\in X:[x]\in S\}=\bigcup _{s\in S}q^{-1}(s).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb41290ccaf125478c66d394f1fde27a834edcb0)
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,
![{\displaystyle \tau _{Y}=\left\{U\subseteq Y:\{x\in X:[x]\in U\}\in \tau _{X}\right\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95aebc8bdd4bb58d06d56df200c1f896f08e6d85)
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) 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.
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.
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.