Exercises:Mond - Topology - 1/Question 8

Section B

Question 8

Suppose that [ilmath]f:X\rightarrow Y[/ilmath] is a continuous function that is also surjective, and let [ilmath]\sim[/ilmath] denote the equivalence relation induced by [ilmath]f[/ilmath] on [ilmath]X[/ilmath].

Show that if [ilmath]X[/ilmath] is compact and [ilmath]Y[/ilmath] is Hausdorff then [ilmath]\frac{X}{\sim} [/ilmath] is homeomorphic to [ilmath]Y[/ilmath].

Which theorem of group theory does this resemble?

Proof

Recall the result of question 5:

• "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"

Using that we instantly obtain:

• [ilmath]\bar{f}:\frac{X}{\sim}\rightarrow Y[/ilmath] that is a continuous bijection
  • We must show this is a homeomorphism.

Suppose [ilmath]X[/ilmath] is compact, then [ilmath]\frac{X}{\sim} [/ilmath] is compact also as the image of a compact set is compact and [ilmath]\pi:X\rightarrow\frac{X}{\sim} [/ilmath] is continuous (see quotient topology for more details)

We then apply the "compact-to-Hausdorff theorem" which shows us that [ilmath]\bar{f} [/ilmath] is actually a homeomorphism

This is similar to the first group isomorphism theorem in that it uses passing to the quotient through morphisms in the category to yield an isomorphism (which are called homeomorphisms in TOP)

Note that this is extremely similar (and is not much more than) the group homomorphism theorem, which is basically the first iso. theorem in the case of [ilmath]f[/ilmath] being surjective.