Quotient topology

Definition

Let $(X, \tau)$ be a topological space and let $\sim$ be an equivalence relation on $X$. Let $q \colon X \to X / {\sim}$ be the map to the quotient set that sends $x$ to its class. The quotient topology on $X / {\sim}$ is $$\{ V \subseteq X / {\sim} : q^{-1}(V) \in \tau \}.$$

Remarks

It is the finest topology on $X / {\sim}$ for which $q$ is continuous. A preimage $q^{-1}(V)$ is a union of classes. Thus $V$ is open if and only if the union of the classes in $V$ is open in $X$. Quotients need not keep good properties: a quotient of a Hausdorff space need not be Hausdorff. Examples are the circle, the torus and the Möbius strip.

Explanations

L1 human

You make a quotient space by gluing the points of each class into one point. For example, gluing the two ends of $[0, 1]$ gives a circle.