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The \(G_{\delta}\)-topology and K-analytic spaces without perfect compact sets. (English) Zbl 0727.54017

Let X be Tychonoff. An algebra A on X is defined to be a subring of C(X) separating points and closed sets, containing the real constants, and closed under uniform convergence and inversion. For any algebra A on X we obtain the Baire functions B(A) generated by A as \(\cup \{B_{\sigma}(A):\;\sigma <\omega_ 1\},\) where \(B_{\alpha}(A)\) is defined inductively by \(B_ 0(A)=A\), \(B_{\alpha}(A)\) is the space of pointwise limits of functions in \(\cup \{B_{\sigma}(A):\;\sigma <\alpha \}.\) In the first part of his paper, the author shows that if each function in A has countable image then each function in B(A) has countable image, thereby answering a question of R. Levy, and M. D. Rice [Colloq. Math. 44, 227-240 (1981; Zbl 0496.54034)]. In the second part of this paper some new characterizations are given of K- analytic spaces without compact perfect subsets. E.g., a K-analytic X does not contain a compact perfect subset iff every covering of X consisting of \(G_{\delta}\)-sets contains a countable subcover. As a consequence of his results the author proves that a perfectly normal K- analytic space either contains a compact perfect subset or is the countable union of dispersed compact subspaces; this does not hold for arbitrary K-analytic spaces, but does hold for arbitrary K-Lusin spaces [J. E. Jayne, Ann. Inst. Fourier 24(1974), No.4, 47-76 (1975; Zbl 0287.46031)].

MSC:

54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
54C50 Topology of special sets defined by functions
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