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
Contents
- 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 [ilmath](X,\mathcal{ J })[/ilmath] and [ilmath](Y,\mathcal{ K })[/ilmath] be topological spaces and let [ilmath]f:X\rightarrow Y[/ilmath] be a surjective continuous map. Then [ilmath]f[/ilmath] can be factored through the canonical projection of the equivalence relation induced by [ilmath]f[/ilmath] to yield a continuous bijection^{[Note 1]}, [ilmath]\bar{f}:\frac{X}{\sim}\rightarrow Y[/ilmath]^{[1]}.
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 [ilmath]\bar{f}:\frac{X}{\sim}\rightarrow Y[/ilmath] is injective and continuous.
Recall from passing to the quotient that if [ilmath]f[/ilmath] is surjective then so is [ilmath]\bar{f} [/ilmath] - we apply that here (we know we can factor as factoring is how we got [ilmath]\bar{f} [/ilmath] in the first place), thus [ilmath]\bar{f} [/ilmath] is surjective!
A surjective injection is of course called a bijection.
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Notes
- ↑ This may not be a homeomorphism (a topological isomorphism however! That would require its inverse was also continuous
References