Universal Property of Quotient Space

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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Also see

• Universal Property of Quotient of Topological Group