Weakly Baire spaces.

*(English)*Zbl 0665.54019Pick a decent \(\sigma\)-ideal I, e.g., of all countable or all first category sets. Call a space X an I-Baire if no nonempty open, dense-in- itself (if necessary) set belongs to I. These ideas have been studied on a piecemeal basis by various authors, see for example J. Kaniewski and the reviewer, Proc. Am. Math. Soc. 98, 324-328 (1986; Zbl 0606.54008), esp. p. 328.

The authors consider “weakly Baire spaces” - that is I-Baire spaces, I being the \(\sigma\)-ideal of all countable sets. This paper contains interesting results pertaining to topological operations on these spaces, such as sums, products or hyperspaces. Two questions are posed. One of them has been recently answered by A. Szymański (unpublished); namely if X is a \(T_ 1\) quasi-regular space, then the following are equivalent: (1) Each non-empty open subset of \(2^ X\) \((=the\) space of all closed subsets of X with Vietoris topology has cardinality at least \(2^{\aleph_ 0}\). (2) \(2^ X\) is weakly Baire. (3) X is dense-in- itself.

The authors consider “weakly Baire spaces” - that is I-Baire spaces, I being the \(\sigma\)-ideal of all countable sets. This paper contains interesting results pertaining to topological operations on these spaces, such as sums, products or hyperspaces. Two questions are posed. One of them has been recently answered by A. Szymański (unpublished); namely if X is a \(T_ 1\) quasi-regular space, then the following are equivalent: (1) Each non-empty open subset of \(2^ X\) \((=the\) space of all closed subsets of X with Vietoris topology has cardinality at least \(2^{\aleph_ 0}\). (2) \(2^ X\) is weakly Baire. (3) X is dense-in- itself.

Reviewer: Z.Piotrowski