SCHWEITZER 2004 Problem 1

The Lindelöf number $L(X)$ of a topological space $X$ is the least infinite cardinal $\lambda$ with the property that every open covering of $X$ has a subcovering of cardinality at most $\lambda$ . Prove that if evert non-countably infinite subset of a first countable space $X$ has a point of condensation, then $L(X)=\sup L(A)$ , where $A$ runs over the separable closed subspaces of $X$ . (A point of condensation of a subset $H\subseteq X$ is a point $x\in X$ such that any neighbourhood of $x$ intersects $H$ in a non-countably infinite set.)