Universal Property of Quotient Space
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Theorem
Let $X$ and $Y$ be topological spaces.
Let $\sim$ be an equivalence relation on $X$.
Let $\pi : X \to X / {\sim}$ be the quotient mapping.
Let $f : X \to Y$ be continuous and $\sim$-invariant.
Then there exists a unique continuous map $\overline f : X / {\sim} \to Y$ such that $f = \overline f \circ \pi$.
Proof
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