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Čech analytic and almost \(K\)-descriptive spaces. (English) Zbl 0806.54030
Recall that a topological space \(X\) is Čech-complete if it is a \(G_ \delta\) in some compact (Hausdorff) space, and Čech-analytic if it can be obtained from the Borel sets of \(X\) by the Suslin operation (equivalently, Čech-analytic spaces are the completely regular projections of Čech-complete spaces along a separable metrizable factor). A family \({\mathcal U}\) in \(X\) is called \(\text{sb}_ d\)-\(\sigma\)- decomposable if \(U = \bigcup\{U_ n : n < \omega\}\) for each \(U \in {\mathcal U}\) such that each family \(\{U_ n : U \in {\mathcal U}\}\) is a disjoint family with a scattered base (equivalently, if \({\mathcal U}\) is point-countable with \(\sigma\)-scattered base). Finally, a space \(X\) is almost \(K\)-descriptive (resp. almost descriptive) if there is a completely metrizable \(M\) and an upper semi-continuous (resp. continuous) compact-valued map \(f:M \to X\) which preserves \(\text{sb}_ d\)- \(\sigma\)-decomposable families.
The authors show that every Čech-analytic space is almost \(K\)- descriptive, and that almost \(K\)-descriptive and Čech-analytic coincide with each other (and with analyticity) in metric spaces. Furthermore, they show that the class of almost \(K\)-descriptive spaces shares various properties with the Čech-analytic spaces: it is closed under countable unions and intersections, the Suslin operation, closed subspaces, and open subspaces; and an almost \(K\)-descriptive completely regular space is \(\sigma\)-scattered or contains a compact perfect set. Finally, it is shown that almost \(K\)-descriptive and almost descriptive coincide for subspaces of Banach spaces with the weak topology.

MSC:
54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
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References:
[1] G. Fodor: On stationary sets and regressive functions. Acta Sci. Math. (Szeged) 27, 105-110. · Zbl 0199.02102
[2] D.H. Fremlin: Čech-analytic spaces.
[3] Z. Frolík: Topologically complete spaces. Comm. Math. Univ. Carol. 1 (1960), 3-15. · Zbl 0100.18702
[4] Z. Frolík: On the topological product of paracompact spaces. Bull. Acad. Polon. Sci. Ser. Sci. Math. Astr. Phys. VIII (1960), 747-750. · Zbl 0099.38601
[5] Z. Frolík: Absolute Borel and Souslin sets. Pac. J. Math. 32 (1970), no. 3, 663-683. · Zbl 0215.24002
[6] Z. Frolík and P. Holický: Analytic and Luzin spaces (non-separable case). Topology and its Appl. 19 (1985), 129-156. · Zbl 0579.54026
[7] Z. Frolík and P. Holický: Applications of Luzinian separation priciples (non-separable case). Fund. math. CXVII (1983), 165-185. · Zbl 0543.54035
[8] G. Gruenhage and J. Pelant: Analytic spaces and paracompactness of \(X^2\backslash \Delta \). Topology and its Appl. 28 (1988), 11-15. · Zbl 0636.54025
[9] R.W. Hansell: Descriptive sets and the topology of nonseparable Banach spaces. · Zbl 0982.46012
[10] R.W. Hansell: On characterizing nonseparable analytic and extended Borel sets as types of continuous images. Proc. London Math. Soc. 28 (1974), no. 3, 683-699. · Zbl 0313.54044
[11] J.E. Jayne, I. Namioka and C.A. Rogers: Properties like the Radon-Nikodým property.
[12] G. Koumoullis: Topological spaces containing compact perfect sets and Prohorov spaces. Topology and its Appl. 21 (1985), 59-71. · Zbl 0574.54041
[13] I. Namioka: Radon-Nikodým compact spaces and fragmentability. Mathematika 34 (1987), 258-281. · Zbl 0654.46017
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