Summary
In this paper we study topological properties of Baire sets in various classes of spaces. The main results state that a Baire set in a realcompact space is realcompact; a Baire set in a topologically complete space is topologically complete; and that a pseudocompact Baire set in any topological space is a zero-set. As a consequence, we obtain new characterizations of realcompact and pseudocompact spaces in terms of Baire sets of their Stone-Čech compactifications. (Lorch in [3] using a different method has obtained either implicitly or explicitly the same results concerning Baire sets in realcompact spaces.) The basic tools used for these proofs are first, the notions of anr-compactification andr-embedding (see below for definitions), which have also been defined and used independently byMrówka in [4]; second, the idea included in the proof of the theorem: “Every compact Baire set is aG δ ” as given inHalmos' text on measure theory [2; Section 51, theorem D].
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References
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The author wishes to thank Professor W. W.Comfort for his valuable advice in the preparation of this paper.
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Negrepontis, S. Baire sets in topological spaces. Arch. Math 18, 603–608 (1967). https://doi.org/10.1007/BF01898869
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DOI: https://doi.org/10.1007/BF01898869