On spaces of Baire \(I\) functions over \(K\)-analytic spaces. (English. Russian original) Zbl 0790.54016
Math. Notes 52, No. 3, 953-959 (1992); translation from Mat. Zametki 52, No. 3, 108-116 (1992).
Let \(X\) be an \(K\)-analytic space and let \(B_ 1(X)\) be the space of all real Baire 1 functions over \(X\) equipped with the pointwise convergence topology. Suppose that \(F\subset B_ 1(X)\) is a relatively countable compact space. Then the closure of \(F\) is a strongly countable compact Fréchet-Urysohn space; and if \(F\) is \(\aleph_ 1\)-compact then \(F\) is a bicompactum. Moreover, if \(X\) is regular then the following conditions are equivalent: (1) The closure of every relatively countable compact set in \(B_ 1(X)\) is bicompact; (2) \(B_ 1(X)\) does not contain a closed subset homeomorphic to \(\omega_ 1\); (3) \(B_ 1(X)\) does not contain a subset homeomorphic to \(\omega_ 1\); (4) \(X\) is paracompact.
Reviewer: Z.Grande (Bydgoszcz)
MSC:
| 54C30 | Real-valued functions in general topology |
| 54H05 | Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets) |
| 54D15 | Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.) |
| 54D20 | Noncompact covering properties (paracompact, Lindelöf, etc.) |
Keywords:
\(K\)-analytic space; Baire 1 functions; pointwise convergence topology; strongly countable compact Fréchet-Urysohn spaceReferences:
| [1] | I. Bourgain, D. H. Fremlin, and M. Talagrand, ?Pointwise compact sets of Bairemeasurable functions,? Amer. J. Math.,100, No. 4, 845-886 (1978). · Zbl 0413.54016 · doi:10.2307/2373913 |
| [2] | A. V. Arkhangel’skii, ?Function spaces in the pointwise convergence topology and compacta,? Usp. Mat. Nauk,39, No. 5, 11-50 (1984). |
| [3] | K. Kuratowski, TopologyVol. 1 [Russian translation], Mir, Moscow (1966). |
| [4] | M. Laczkovich, ?Baire I functions,? Real. Anal. Exch.,9, No. 1, 15-28 (1983-84). |
| [5] | G. Choquet, ?Borel and analytic sets in topological spaces,? C. R. Acad. Sci. Paris,232, 2174-2176 (1951). |
| [6] | I. E. Iayne, ?Structure of analytic Hausdorff spaces,? Mathematika,23, No. 2, 208-211 (1976). · Zbl 0348.54031 · doi:10.1112/S0025579300008809 |
| [7] | Z. Frolik, ?On the topological product of paracompact spaces,? Bull. Acad. Pol. Sci. Ser. Math.,8, No. 11-12, 747-750 (1960). |
| [8] | A. V. Arkhangel’skii, ?Suslin’s number and cardinality. Characters of points in sequential bicompacta,? Dokl. Akad. Nauk SSSR,192, No. 2, 255-258 (1970). |
| [9] | I. Bourgain, ?Some remarks on compact sets of first Baire class,? Bull. Soc. Math. Belg.,30, No. 1, 3-10 (1978). · Zbl 0414.54011 |
| [10] | A. V. Arkhangel’skii and V. I. Ponomarev, Foundations of General Topology in Problems and Exercises [in Russian], Nauka, Moscow (1974). |
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