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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.

54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
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