How to solve this question?
Let $X$ be a Hausdorff space and $\{K_n\}$ a familly of decreasing compact non empty compact sets, Let $K=\bigcap_{n\in\mathbb{N}} K_n$ and let $\Omega$ be an open set $$K\subset \Omega\Rightarrow \exists n_0\in\mathbb{N},\forall n\in \mathbb{N}, K_n\subset \Omega$$.
If I suppose by contradiction that $K\subset \Omega$ and $$\forall n_0\in\mathbb{N},\exists n\in\mathbb{N}, n\geq n_0 ~\text{and}~ K_n\not\subset \Omega$$.
That is
$$\forall n_0\in\mathbb{N},\exists n\in\mathbb{N}, n\geq n_0, \exists x\in K_n ~\text{and}~ x\in E\setminus \Omega$$
How to continue? Is there any other method?
If not, then we have an open set $\Omega$ containing $K$ such that $$\forall n \in \mathbb{N}: K_n \nsubseteq \Omega$$
and then we can thus define $C_n = K_n\setminus \Omega = K_n \cap( X\setminus \Omega)$ and all $C_n$ are non-empty (by the non-inclusion) and decreasing (as the $K_n$ are) and compact, $C_n$ being closed in $K_n$.
But then $\cap_n C_n \neq \emptyset$ by a standard fact on families of closed sets (by Hausdorffness of $X$) with the f.i.p. in a compact space ($C_0$). But any $p \in \cap_n C_n$ is in all $K_n$, hence in $K$, but not in $\Omega$ contradiction.