If a surjective continuous map is factored through the canonical projection of the equivalence relation induced by that map then the yielded map is a continuous bijection

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Note: "Factoring a continuous map through the projection of an equivalence relation induced by that map yields an injective continuous map" is an important precursor theorem

Statement

Let $(X,\mathcal{ J })$ and $(Y,\mathcal{ K })$ be topological spaces and let $f:X\rightarrow Y$ be a surjective continuous map. Then $f$ can be factored through the canonical projection of the equivalence relation induced by $f$ to yield a continuous bijection, $\bar{f}:\frac{X}{\sim}\rightarrow Y$.

Proof

We know already (from: "Factoring a continuous map through the projection of an equivalence relation induced by that map yields an injective continuous map" that $\bar{f}:\frac{X}{\sim}\rightarrow Y$ is injective and continuous.

Recall from passing to the quotient that if $f$ is surjective then so is $\bar{f}$ - we apply that here (we know we can factor as factoring is how we got $\bar{f}$ in the first place), thus $\bar{f}$ is surjective!

A surjective injection is of course called a bijection.

Grade: C

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