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 $\pi :X⟶X^{*}$ 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 $p:X⟶Y$ 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

$$x\sim x^{\prime }\iff p(x)=p(x^{\prime }).$$

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.