Baire spaces, \( \mathrm{G}_\delta \)’s, and directed complete spaces. (English) Zbl 1476.54028
This paper is a thorough survey of developments that have their origins in the Baire Category Theorem, see [R. Baire, Annali di Mat. (3) 3, 1–123 (1899; JFM 30.0359.01)], where on page 65 we find the notions of sets “du première catégorie et du deuxième catégorie”. The theorem also appears, but less prominently, in [W. F. Osgood, Am. J. Math. 19, 155–190 (1897; JFM 28.0221.01)].
The fact that this theorem holds in the two diverse classes of completely metrizable spaces and locally compact spaces, respectively, has spurred much research into finding simultaneous generalizations of these two notions where the theorem continues to hold. One obvious class is that of the Baire spaces, defined by “the Baire Category Theorem is valid for this space”, which is a rather unsatisfactory characterization.
The paper offers a comprehensive list of proposed types of spaces that all sit between the two classes above on one side and the Baire spaces on the other. The first half of the paper provides and excellent introduction to the theorem for undergraduate students, which one could supplement with a few examples of its use, such as showing that almost all continuous functions on the interval \([0,1]\) are nowhere differentiable. The second half (from section 8 onward) contains more recent material concerning subcompactness and various spaces characterized by winning strategies for players of topological games.
Worth perusing even now and then, also for the various open questions in this area.
The fact that this theorem holds in the two diverse classes of completely metrizable spaces and locally compact spaces, respectively, has spurred much research into finding simultaneous generalizations of these two notions where the theorem continues to hold. One obvious class is that of the Baire spaces, defined by “the Baire Category Theorem is valid for this space”, which is a rather unsatisfactory characterization.
The paper offers a comprehensive list of proposed types of spaces that all sit between the two classes above on one side and the Baire spaces on the other. The first half of the paper provides and excellent introduction to the theorem for undergraduate students, which one could supplement with a few examples of its use, such as showing that almost all continuous functions on the interval \([0,1]\) are nowhere differentiable. The second half (from section 8 onward) contains more recent material concerning subcompactness and various spaces characterized by winning strategies for players of topological games.
Worth perusing even now and then, also for the various open questions in this area.
Reviewer: K. P. Hart (Delft)
MSC:
| 54E52 | Baire category, Baire spaces |
| 54G20 | Counterexamples in general topology |
| 54D70 | Base properties of topological spaces |
| 91A44 | Games involving topology, set theory, or logic |
| 54-02 | Research exposition (monographs, survey articles) pertaining to general topology |
Keywords:
Baire Category Theorem; Baire space; subcompact; \(G_\delta\)-set; Banach-Mazur game; Choquet game; directed complete; \(\alpha\)-favourableReferences:
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