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}$ .