help me in Prove

**nice rose** · May 9th 2010, 06:58 AM

Prove

1) $X$ is a regular space, $A$ compact set and $B$ closed set,
$A \bigcap B = {\O}$
Then there is open sets U and V exist, such that
$A\subseteq U, \ B \subseteq V,\ A \bigcap B = {\O}$

2) $X$ is compact and hausdorff space and $f : X \rightarrow X$ is continuous function
prove there is a closed set non empty such that $f(A)= A$

**roninpro** · May 9th 2010, 08:09 AM

For part two, define a sequence of sets in the following way: $A_0=X, A_1=f(A_0), A_2=f(A_1), A_3=f(A_2),\ldots$ . Since $f:X\to X$ is continuous and takes a compact space to a Hausdorff space, the closed map lemma applies; we can say that each $A_i$ is a closed set. Let $A=\bigcap A_i$ . Note that $A$ is closed and $f(A)=A$ . To complete the proof, it necessary to show that $A$ is nonempty. I can't see how to do that at the moment, so hopefully somebody else can fill in the gap.

I hope that this gives you some ideas, at any rate.

**Drexel28** · May 9th 2010, 10:28 AM

Originally Posted by nice rose

Prove

1) $X$ is a regular space, $A$ compact set and $B$ closed set,
$A \bigcap B = {\O}$
Then there is open sets U and V exist, such that
$A\subseteq U, \ B \subseteq V,\ A \bigcap B = {\O}$

Oh come on!

Hint:

Spoiler: