# Compactness via closed sets

Do you know why compactness of a set, in a metric space, can’t be expressed in terms of closed sets?

That is, the next statement is false: Let $$(X,d)$$ a metric space and $$A\subseteq X$$ such that every closed cover of $$A$$ has a finite subcover. Then $$A$$ is compact (in the usual sense).

I’m trying to find a non compact set $$A$$ on a metric space $$X$$ such that, whenever $$\{F_\lambda\}$$ is a collection of closed sets such that $$A\subseteq\cup_{\lambda}F_\lambda$$, there are $$\lambda_1,…,\lambda_n$$ with $$A\subseteq\cup_{i=1}^n F_{\lambda_i}$$.