## 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)
Full Text:

### 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.