A strengthening of the Cech-Pospišil theorem

István Juhász and Zoltán Szentmiklóssy

We prove the following result: If in a compact space $X$ there is a $\lambda$-branching family of closed sets then $X$ cannot be covered by fewer than $\lambda$ many discrete subspaces. (A family of sets $\mathcal{F}$ is $\lambda$-branching iff $\vert\mathcal{F}\vert < \lambda$ but one can form $\lambda$ many pairwise disjoint intersections of subfamilies of $\mathcal{F}.$) The proof is based on a recent, still unpublished, lemma of G. Gruenhage.

As a consequence, we obtain the following strengthening of the well-known Cech-Pospišil theorem: If $X$ a is compact $T_2$ space such that all points $x \in X$ have character $\chi(x,X) \ge \kappa$ then $X$ cannot be covered by fewer than $2^\kappa$ many discrete subspaces.

Key words and phrases: compact space, discrete subspace, covering, Cech - Pospišil theorem

2000 Mathematics Subject Classification: 54A25, 54B05

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