4.11 Quotient Topologies
Definition 4.11.1 (Saturated). Let $X, Y$ be sets, $f: X \to Y$ be surjective, and $E \subset X$, then $E$ is saturated with respect to $f$ if $E = f^{-1}(f(E))$.
Definition 4.11.2 (Quotient Map). Let $X, Y$ be topological spaces and $\pi: X \to Y$ be a surjective map, then the following are equivalent:
For any $U \subset Y$, $U$ is open if and only if $\pi^{-1}(U)$ is open.
$\pi \in C(X; Y)$, and for any $U \subset X$ saturated and open, $\pi(U)$ is open.
If the above holds, then $\pi$ is a quotient map.
Proof. $(1) \Rightarrow (2)$: Let $U \subset Y$ be open, then $\pi^{-1}(U)$ is open, so $f \in C(X; Y)$. If $V \subset X$ is saturated and open, then $\pi(V) = \pi(\pi^{-1}(\pi(V)))$ is open.
$(2) \Rightarrow (1)$: Let $U \subset Y$ be open, then $\pi^{-1}(U)$ is open by continuity. If $V \subset Y$ and $\pi^{-1}(V)$ is open, then $V = \pi(\pi^{-1}(V))$ is open.$\square$
Definition 4.11.3 (Quotient Space). Let $X$ be a topological space, $\sim$ be an equivalence relation on $X$, then there exists $(\td X, \pi)$ such that:
$\td X$ is a topological space with ground set $X/\sim$.
$\pi \in C(X; \td X)$.
$\pi$ is constant on each equivalence class of $\sim$.
For any pair $(Y, f)$ satisfying (1), (2), and (3), there exists a unique $\td f \in C(\td X; Y)$ such that the following diagram commutes:
\[\xymatrix{ X \ar@{->}[d]_{\pi} \ar@{->}[rd]^{f} & \\ \td X \ar@{->}[r]_{\tilde f} & Y }\]$\pi$ is a quotient map.
The space $(\td X, \pi)$ is the quotient of $X$ by $\sim$.
Proof. Let $\td X = X/\sim$. For each $U \subset \td X$, define $U$ to be open if and only if $\pi^{-1}(U) \subset X$ is open, then $(\td X, \pi)$ satisfies (1), (2), (3), and (5).
(U): Since $f$ is constant on each equivalence class of $\sim$, there exists $\td f: \td X \to Y$ such that the diagram commutes. For any $U \subset Y$ open, $\td f^{-1}(U) = \pi(f^{-1}(U))$ is saturated with respect to $\pi$, so $\td f^{-1}(U)$ is open.$\square$