Quotient space (topology) on arvau

Wikipedia / 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 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

be a topological space be an equivalence relation The quotient set is the set of equivalence classes

defines a canonical surjection

The quotient space equipped with the quotient topology, whose open sets are those subsets whose preimage is open if and only if is closed

The quotient topology is the final topology

Quotient map

is a quotient map (sometimes called an identification map) if it is surjective is equipped with the final topology is open (resp. closed). Every quotient map is continuous but not every continuous map is a quotient map.

Saturated sets

is called saturated establishes a one-to-one correspondence is a quotient map if and only if for every that are is open in is open in

is a topology is a quotient map (respectively, continuous

Quotient space of fibers characterization

Given an equivalence relation denote the equivalence class that sends points to their equivalence classes) is called . It is a surjective map into a quotient map. Up to a homeomorphism, this construction is representative of all quotient spaces; the precise meaning of this is now explained.

) is a singleton set defines a bijection is a homeomorphism and its inverse are continuous).

Related definitions

A is also a quotient map. There exist quotient maps that are not hereditarily quotient.

Examples

Gluing).
is homeomorphic to the sphere
Adjunction space is a subspace The 2-sphere is then homeomorphic to a closed disc
of real numbers if and only if is an integer is homeomorphic to the unit circle

A generalization of the previous example is the following: Suppose a topological group acts 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

Note via addition, then the quotient (which is identifiable) is a countably infinite bouquet of circles
This next example shows that it is in general is a quotient map then every convergent sequence (respectively, every convergent net has a lift) to a convergent sequence (or convergent net) and a homeomorphism). This counterexample can be generalized to nets be any directed set

Properties

is continuous.

is characterized by the following universal property In other words, the following diagram commutes:

descends to the quotient 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.

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

Separation properties

In general, quotient spaces are ill-behaved with respect to separation axioms

is a T1 space to a single point is homeomorphic to Sierpiński space, which is not T1.

to be a Hausdorff space must be closed in the product space is not regular

If the quotient map is open

Some separation properties are preserved under certain conditions. In particular,

is a T4 space (i.e., normal Hausdorff) and the quotient map is closed is also T4.
is a T6 space is also T6.

Connectedness

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.

Compactness

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.

Dimension

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.