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

Volume 141 / 2015

Samuel G. da Silva Colloquium Mathematicum 141 (2015), 199-208 MSC: Primary 54D45, 54A25, 03E05; Secondary 54A35, 03E65, 03E17. DOI: 10.4064/cm141-2-5


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.


  • Samuel G. da SilvaInstituto de Matemática
    Universidade Federal da Bahia
    Av. Adhemar de Barros, S/N, Ondina
    CEP 40170-110, Salvador, BA, Brazil

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