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 $f = \overline f \circ \pi$.

Also see

 * Universal Property of Quotient of Topological Group