Quotient topology

For an equivalence relation $R$ on a topological space $X$, the quotient topology on the set $X/R$ declares $U\subseteq X/R$ open exactly when its inverse image under the projection $q:X\to X/R$ is open in $X$.

A quotient topological space identifies points of a topological space according to an equivalence relation and equips the set of equivalence classes with the quotient topology. A subset is open exactly when its inverse image under the canonical quotient map is open. Even a Hausdorff space can have a non-Hausdorff quotient; identifying real numbers differing by a rational number gives a multi-point quotient with the indiscrete topology.

A quotient map is a surjective continuous map $q:X\to Y$ for which $U\subseteq Y$ is open exactly when $q^{-1}(U)$ is open.

If $q:X\to X/R$ is the quotient map and a continuous map $f:X\to Y$ is constant on each equivalence class, there is a unique continuous map $F:X/R\to Y$ satisfying $Fq=f$. It is given by $F([x])=f(x)$; continuity follows becausefor every open $U\subseteq Y$.

Identify $x,y\in \mathbb{R}$ when $x-y\in \mathbb{Q}$. Every nonempty saturated open subset of $\mathbb{R}$ meets every other one because rational translates of any open interval meet any prescribed nonempty open interval. The quotient $\mathbb{R}/\mathbb{Q}$ has multiple points but no two nonempty disjoint open subsets, so it is not Hausdorff.

Identify the vertical sides of $[0,1{]}^2$, then collapse each horizontal side separately to a point. The resulting quotient is homeomorphic to $S^2$ throughEquivalently, the quotient is the suspension of the circle.