quotient space

Let $X$ be a topological space, and let $\sim$ be an equivalence relation on $X$ . Write $X^*$ for the set of equivalence classes of $X$ under $\sim$ . The quotient topology on $X^*$ is the topology whose open sets are the subsets $U \subset X^*$ such that $$ \bigcup U \subset X $$ is an open subset of $X$ . The space $X^*$ is called the quotient space of the space $X$ with respect to $\sim$ . It is often written $X/\sim$ .

The projection map $\pi: X \longrightarrow X^*$ which sends each element of $X$ to its equivalence class is always a continuous map. In fact, the map $\pi$ satisfies the stronger property that a subset $U$ of $X^*$ is open if and only if the subset $\pi^{-1}(U)$ of $X$ is open. In general, any surjective map $p: X \longrightarrow Y$ that satisfies this stronger property is called a quotient map, and given such a quotient map, the space $Y$ is always homeomorphic to the quotient space of $X$ under the equivalence relation $$ x \sim x' \iff p(x) = p(x'). $$

As a set, the construction of a quotient space collapses each of the equivalence classes of $\sim$ to a single point. The topology on the quotient space is then chosen to be the strongest topology such that the projection map $\pi$ is continuous.

For $A \subset X$ , one often writes $X/A$ for the quotient space obtained by identifying all the points of $A$ with each other.