Quotient topology

The quotient topology is the canonical way of defining a topology on a Algebraic quotient, as defined by an Equivalence relation or projection. Let be a topological space, and ¹ be a surjective function. The quotient topology on is the finest topology for which is continuous.² #m/def/topology

Further characterisations

Universal property

For every topological space and , then is continuous iff . #m/thm/topology

Proof

First we will prove that the quotient topology as characterised above satisfies the universal property. Let be a topological space, be a surjective function, and be endowed with the quotient topology . Let be some topological space, and let be a function. If is continuous, then so is the composition of continuous functions. Now suppose is continuous, and let . Then whence . Thus is continuous. Therefore is continuous iff is continuous.

Now let be a topology on satisfying the universal property. In particular, let and . Then since is continuous so is , wherefore is finer than Now let and . Since is continuous, so too is . But is the finest topology for which is continuous, so .

Further terminology

An equivalence relation is called a Closed equivalence relation iff it is closed regarded as a subset of

Properties

From the universal property, a function is continuous iff is continuous and constant for the fibres of .
A function is said to factor through iff it is constant for fibres of .

Spaces constructed as quotients

Unit circle topology as defined by with
Möbius strip, Klein bottle, and other shapes constructed using a Fundamental polygon
Projective space

#state/tidy | #lang/en | #SemBr

where is often constructed as the fibres of , which is precisely the Algebraic quotientient]] ↩
2020, Topology: A categorical approach, pp. 28–29 ↩