The quotient topology

Definition 397.1(Definition).

Let $X$ and $Y$ be topological spaces, let $p:X\to Y$ be a surjective map. The map $p$ is said to be a quotient map provided a subset $U$ of $Y$ is open in $Y$ iff $p^{-1}(U)$ is open in $X$.

We say that a subset $C$ of $X$ is saturated with respective to a surjective map $p:X\to Y$ if $C$ is a union of sets of the form $p^{-1}(\{y\})$ for $y\in Y$. Thus, $p$ is a quotient map iff it is continuous and maps saturated open sets of $X$ to open sets of $Y$.

Definition 397.2(Definition).

If $X$ is a space and $A$ is a set and if $p:X\to A$ is a surjective map, then there exists exactly one topology $\mathcal{T}$ on $A$ relative to which $p$ is a quotient map; it is called the quotient topology induced by $p$.

Definition 397.3(Definition).

Let $X$ be a topological space, and let $X^{\ast }$ be a partition of $X$ into disjoint subsets whose union is $X$. Let $p:X\to X^{\ast }$ be the surjective map that carries each point of $X$ to the element of $X^{\ast }$ containing it. In the quotient topology induced by $p$, the space $X^{\ast }$ is called a quotient space of $X$.

Proposition 397.4(Proposition).

Let $p:X\to Y$ be a quotient map; let $A$ be a subspace of $X$ that is saturated with respect to $p$. Let $q:A\to p(A)$ be the map obtained by restricting $p$.

1. If $A$ is either open or closed in $X$, then $q$ is a quotient map.
2. If $p$ is either an open map or a closed map, then $q$ is a quotient map.