SCHWEITZER 1981 Problem 9

Let $n \geq 2$ be an integer, and let $X$ be a connected Hausdorff space such that every point of $X$ has a neighborhood homeomorphic to the Euclidean space $\mathbb{R}^n$ . Suppose that any discrete (not necessarily closed ) subspace $D$ of $X$ can be covered by a family of pairwise disjoint, open sets of $X$ so that each of these open sets contains precisely one element of $D$ . Prove that $X$ is a union of at most $\aleph_1$ compact subspaces.

Z. Balogh