On the extent of separable, locally compact, selectively $(a)$-spaces
Volume 141 / 2015
Colloquium Mathematicum 141 (2015), 199-208
MSC: Primary 54D45, 54A25, 03E05; Secondary 54A35, 03E65, 03E17.
DOI: 10.4064/cm141-2-5
Abstract
The author has recently shown (2014) that separable, selectively $(a)$-spaces cannot include closed discrete subsets of size $\mathfrak {c}$. It follows that, assuming $\mathbf {CH}$, separable selectively $(a)$-spaces necessarily have countable extent. However, in the same paper it is shown that the weaker hypothesis ‶$2^{\aleph _0} < 2^{\aleph _1}$″ is not enough to ensure the countability of all closed discrete subsets of such spaces. In this paper we show that if one adds the hypothesis of local compactness, a specific effective (i.e., Borel) parametrized weak diamond principle implies countable extent in this context.
