I am supposed to solve this question:
1-But I have a difficulty in understanding the finite intersection property mentioned in it. could anyone explain this property for me by a concrete example?
2- Also how this leads leads that $A$ is a compact set, any hints will be appreciated.
The complements of the sets in $\cal F$ are open, and the condition that $$\bigcap_{F\in\cal F}(F\cap A)=\emptyset\tag{1}$$ means that $$\bigcup_{U\in\cal U}(U\cap A)=A\tag{2}$$ where ${\cal U}=\{F^c:F\in\cal F\}$ is the collection of complements of the sets in $\cal F$. The condition $(1)$ is then equivalent to $U\cap A$ are an open cover of $A$.
Compactness means that $(2)$ holding means that it still holds when we replace $\cal U$ by some finite subset. In that case we can replace $\cal F$ in $(1)$ by a finite subset, and that is the finite intersection property.