Gluing_(topology)
Quotient space (topology)
Topological space construction
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 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
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.
- 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
- 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:
- Consider the set
of real numbers with the ordinary topology, and write
if and only if
is an integer. Then the quotient space
is homeomorphic to the unit circle
via the homeomorphism which sends the equivalence class of
to
- A generalization of the previous example is the following: Suppose a topological group
acts 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
- Note: The notation
- This next example shows that it is in general not true that if
is a quotient map then every convergent sequence (respectively, every convergent net) 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 toin
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.
- In general, quotient spaces are ill-behaved with respect to separation axioms. The separation properties of
need not be inherited by
and
may have separation properties not shared by
is a T1 space if and only if every equivalence class of
is closed in
- If the quotient map is open, then
is a Hausdorff space if and only if ~ is a closed subset of the product space
- If a space is connected or path connected, then so are all its quotient spaces.
- A quotient space of a simply connected or contractible space need not share those properties.
- If a space is compact, then so are all its quotient spaces.
- A quotient space of a locally compact space need not be locally compact.
- The topological dimension of a quotient space can be more (as well as less) than the dimension of the original space; space-filling curves provide such examples.
Topology
- Covering space – Type of continuous map in topology
- Disjoint union (topology) – space formed by equipping the disjoint union of the underlying sets with a natural topology called the disjoint union topologyPages displaying wikidata descriptions as a fallback
- Final topology – Finest topology making some functions continuous
- Mapping cone (topology) – topological constructionPages displaying wikidata descriptions as a fallback
- Product space – Topology on Cartesian products of topological spacesPages displaying short descriptions of redirect targets
- Subspace (topology) – Inherited topologyPages displaying short descriptions of redirect targets
- Topological space – Mathematical space with a notion of closeness
Algebra
- Quotient category
- Quotient group – Group obtained by aggregating similar elements of a larger group
- Quotient space (linear algebra) – Vector space consisting of affine subsets
- Mapping cone (homological algebra) – Tool in homological algebra
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