Residuality of families of \({\mathcal F}_\sigma\) sets. (English) Zbl 1098.54028

Saito, Shingo

Real Anal. Exch., Suppl., 29th Summer Symp. Conf. Rep., 97-102 (2005).

The author presents some results on residuality of families of \({\mathcal F}_\sigma\)-sets to be published elsewhere. Let \({\mathcal K}\) denote the family of all closed sets of a space. Denote by \({\mathcal K}^\mathbb{N}\) all sequences in \({\mathcal K}\) (with the product topology), and by \({\mathcal K}^\mathbb{N}_\nearrow\) the subset of all increasing sequences \(\{(K_n)\in{\mathcal K}^\mathbb{N}\mid K_1\subset K_2\subset\cdots\}\). For a family \({\mathcal F}\) of \({\mathcal F}_\sigma\)-sets, the author puts \({\mathcal K}_{\mathcal F}^\mathbb{N}:= \{(K_n)\in{\mathcal K}^\mathbb{N}\mid\bigcup_nK_n\in{\mathcal F}\}\). He defines \({\mathcal F}\) to be \({\mathcal K}^\mathbb{N}\)-residual \(({\mathcal K}^\mathbb{N}_\nearrow\)-residual, resp.) if \({\mathcal K}_{\mathcal F}^\mathbb{N}\) \(({\mathcal K}_{\mathcal F}^\mathbb{N}\cap {\mathcal K}^\mathbb{N}_\nearrow\), resp.) is residual in \({\mathcal K}^\mathbb{N}\) \(({\mathcal K}^\mathbb{N}_\nearrow\), resp.). The main theorem asserts that these two notions are equivalent.

MSC:

54E52	Baire category, Baire spaces
54B20	Hyperspaces in general topology