Welcome to MathOverflow

Question

0

Does the following statement true or not and why? A closed $G_\delta$ of a compact space is zero set of a continuous real-valued function.

flag	asked Mar 15 2012 at 13:41 um Haitham 33●2

Gerald Edgar · Answer 1 · 2012-03-15 14:22:16Z UTC

1

Yes, it is true. In a compact Hausdorff space, "zero set" is the same as "compact $G_\delta$". Since it is a homework type problem, instead of providing a proof I should ask what you have so far...

answered Mar 15 2012 at 14:22

Gerald Edgar
15.2k●1●28●67

	I thought it is true if the space is only normal, but I have found that this result was used by Marty in his paper (m-adic spaces). So does the proof relay on Urysoh's lemma? – um Haitham Mar 15 2012 at 14:35
	thanks for your comment. I forgot that all compact Hausdorff spaces are normal and so this result is true. – um Haitham Mar 15 2012 at 14:53

A compact $ G_\delta$ and zero sets.

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

1 Answer

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.