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 $(𝑋, T 𝑋)$ be a topological space, and $𝜋 : 𝑋 ↠ 𝑆$ ¹ be a surjective function. The quotient topology on $𝑆 ≅ 𝑋 / 𝜋$ is the finest topology for which $𝜋$ is continuous.² #m/def/topology

T 𝜋 = {𝑈 \subseteq 𝑆 : 𝜋 - 1 𝑈 \in T 𝑋}

Further characterisations

Universal property

For every topological space $(𝑍, T 𝑍)$ and $𝑓 : 𝑆 \to 𝑍$ , then $𝑓$ is continuous iff $𝑓 𝜋 : 𝑋 \to 𝑍$ . #m/thm/topology

Proof

First we will prove that the quotient topology as characterised above satisfies the universal property. Let $(𝑋, T 𝑋)$ be a topological space, $𝜋 : 𝑋 ↠ 𝑆$ be a surjective function, and $𝑆$ be endowed with the quotient topology $T 𝜋$ . Let $(𝑍, T 𝑍)$ be some topological space, and let $𝑓 : 𝑆 \to 𝑍$ be a function. If $𝑓$ is continuous, then so is the composition $𝑓 𝜋$ of continuous functions. Now suppose $𝑓 𝜋 : 𝑋 \to 𝑍$ is continuous, and let $𝑈 \in T 𝑍$ . Then $(𝑓 𝜋) - 1 𝑈 = 𝜋 - 1 𝑓 - 1 𝑈 \in T 𝑋$ whence $𝑓 - 1 \in T 𝑆$ . Thus $𝑓$ is continuous. Therefore $𝑓$ is continuous iff $𝑓 𝜋$ is continuous.

Now let $T'$ be a topology on $𝑆$ satisfying the universal property. In particular, let $(𝑍, T 𝑍) = (𝑆, T 𝜋)$ and $𝑓 = i d 𝑆 : 𝑠 \mapsto 𝑠$ . Then since $𝑓 𝜋 = 𝜋$ is continuous so is $𝑓$ , wherefore $T'$ is finer than $T 𝜋$ Now let $(𝑍, T 𝑍) = (𝑆, T')$ and $𝑓 = i d 𝑌$ . Since $i d 𝑌$ is continuous, so too is $i d 𝑌 𝜋 = 𝜋$ . But $T 𝜋$ is the finest topology for which $𝜋$ is continuous, so $T 𝜋 = T'$ .

Further terminology

An equivalence relation is called a Closed equivalence relation iff it is closed regarded as a subset of $𝑋 \times 𝑋$

Properties

From the universal property, a function $𝑓 : 𝑆 \to 𝑍$ is continuous iff $𝜋 : 𝑋 \to 𝑍$ is continuous and constant for the fibres of $𝜋 : 𝑋 ↠ 𝑆$ .
A function $𝑔 : 𝑆 \to 𝑍$ is said to factor through $𝜋$ iff it is constant for fibres of $𝜋 : 𝑋 ↠ 𝑆$ .

Spaces constructed as quotients

Unit circle topology as defined by $[0, 1]$ with $0 \sim 1$
Möbius strip, Klein bottle, and other shapes constructed using a Fundamental polygon
Projective space

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

where $𝑆$ is often constructed as the fibres of $𝜋$ , which is precisely the Algebraic quotientient]] $𝑋 / \sim$ ↩
2020, Topology: A categorical approach, pp. 28–29 ↩