Lemma 5.6.2. Let $X$ be a topological space. Let $Y$ be a set and let $f : X \to Y$ be a surjective map of sets. The quotient topology on $Y$ is the topology characterized by each of the following statements:

it is the strongest topology on $Y$ such that $f$ is continuous,

a subset $V$ of $Y$ is open if and only if $f^{-1}(V)$ is open,

a subset $Z$ of $Y$ is closed if and only if $f^{-1}(Z)$ is closed.

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