# help me in Prove

#### nice rose

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 $$\displaystyle f(A)= A$$

#### roninpro

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

MHF Hall of Honor
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 $$\displaystyle K=\left\{x\in X:f(x)\ne x\right\}=X$$?