On the extent of separable, locally compact, selectively $(a)$-spaces

Samuel G. da Silva¹

¹ Instituto de Matemática Universidade Federal da Bahia Av. Adhemar de Barros, S/N, Ondina CEP 40170-110, Salvador, BA, Brazil

Colloquium Mathematicum, Tome 141 (2015) no. 2, pp. 199-208

Résumé

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.

Détail

DOI : 10.4064/cm141-2-5

Keywords: author has recently shown separable selectively spaces cannot include closed discrete subsets size mathfrak follows assuming mathbf separable selectively spaces necessarily have countable extent however paper shown weaker hypothesis aleph aleph enough ensure countability closed discrete subsets spaces paper adds hypothesis local compactness specific effective borel parametrized weak diamond principle implies countable extent context