Proving Regularity in Y through Closed Continuous Surjective Maps

[guroten]

Homework Statement

Let f : X--> Y be a closed continuous surjective map such that f^(-1)(y) is compact.
Show that if X is regular, so is Y .

The Attempt at a Solution

I'm not sure which piece of info I need to use to start each of these. Any help with the proof would be really appreciated. I already figured out how to show that if X is Hausdorff , so is Y, but I can't figure out how to show it's regular.

 

[Dick]

Pick a closed set C in Y. You know for every element x of f^(-1)(y) that x and the closed set f^(-1)(C) have nonoverlapping neighborhoods. Next use the most basic property of compact sets. That's a hint to get you started.

 

[guroten]

Okay, so I can use compactness to find a finite open covering that is disjoint from the other neighborhood. How can I turn these open sets in X into disjoint open sets in Y? Since the mapping is closed, I have tried to use complements to find the proper disjoint sets, but it is not working.

 

[Dick]

Take the union of the neighborhoods in the finite covering and call it U. Take the intersection of the corresponding neighborhoods of f^(-1)(C) and call it V. Now map those two neighborhoods into Y. What's not working? Oh, I see. So far we only have that f^(-1)(y) and f(-1)(C) are separated in X. Hmmmm.