help me in Prove

Nov 2009
23
0
Prove

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


2) \(\displaystyle X \) is compact and hausdorff space and \(\displaystyle f : X \rightarrow X \) is continuous function
prove there is a closed set non empty such that \(\displaystyle f(A)= A\)
 
Nov 2009
485
184
For part two, define a sequence of sets in the following way: \(\displaystyle A_0=X, A_1=f(A_0), A_2=f(A_1), A_3=f(A_2),\ldots\). Since \(\displaystyle f:X\to X\) is continuous and takes a compact space to a Hausdorff space, the closed map lemma applies; we can say that each \(\displaystyle A_i\) is a closed set. Let \(\displaystyle A=\bigcap A_i\). Note that \(\displaystyle A\) is closed and \(\displaystyle f(A)=A\). To complete the proof, it necessary to show that \(\displaystyle A\) 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

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Nov 2009
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Prove

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

Hint:

For each \(\displaystyle a\in A\) there are disjoint neighborhood \(\displaystyle U_a,V_a\) with \(\displaystyle a\in U_a\) and \(\displaystyle B\subseteq V_a\). Cover \(\displaystyle A\) with the set of all the \(\displaystyle U_a\)'s and procure a finite subcover. What then?

2) \(\displaystyle X \) is compact and hausdorff space and \(\displaystyle f : X \rightarrow X \) is continuous function
prove there is a closed set non empty such that \(\displaystyle f(A)= A\)
Let's see some work

Hint:

What happens if \(\displaystyle K=\left\{x\in X:f(x)\ne x\right\}=X\)?