## $$P$$-filters and hereditary Baire function spaces.(English)Zbl 0969.54016

Summary: We extend the results of S. P. Gul’ko and G. A. Sokolov [ibid. 85, No. 1-3, 137-142 (1998; Zbl 0919.54011)] proving that a filter $$F$$ on $$\omega$$, regarded as a subspace of the Cantor set $$2^\omega$$, is a hereditary Baire space if and only if $$F$$ is a nonmeager (i.e., second category) $$P$$-filter. We also prove related results on hereditary Baire spaces of continuous functions $$C_p(X)$$.

### MSC:

 54C35 Function spaces in general topology 54E52 Baire category, Baire spaces 03E05 Other combinatorial set theory

### Keywords:

nonmeager filter; hereditary Baire space; $$P$$-filter

Zbl 0919.54011
Full Text:

### References:

 [1] Bartoszyński, T.; Judah, H., Set theory: on the structure of the real line, (1995), A.K. Peters Ltd Wellesley, MA · Zbl 0834.04001 [2] Cauty, R.; Dobrowolski, T.; Marciszewski, W., A contribution to the topological classification of the spaces Cp(X), Fund. math., 142, 269-301, (1993) · Zbl 0813.54009 [3] Debs, G., Espaces héréditairement de Baire, Fund. math., 129, 199-206, (1988) · Zbl 0656.54023 [4] Dobrowolski, T.; Marciszewski, W., Classification of function spaces with the pointwise topology determined by a countable dense set, Fund. math., 148, 35-62, (1995) · Zbl 0834.46016 [5] S.P. Gul’ko and G.A. Sokolov, P-points in N* and the spaces of continuous functions, Preprint. [6] Hurewicz, W., Relativ perfekte teile von punktmengen und mengen (A), Fund. math., 12, 78-109, (1928) · JFM 54.0097.06 [7] Marciszewski, W., On analytic and coanalytic function spaces Cp(X), Topology appl., 50, 241-248, (1993) · Zbl 0785.54020 [8] Marciszewski, W.; van Mill, J., An example of tp*-equivalent spaces which are not tp-equivalent, () · Zbl 0918.54013 [9] H. Michalewski, Game-theoretic approach to the hereditary Baire property of Cp(NF), Preprint. · Zbl 0922.54018 [10] van Mill, J., An introduction to βω, (), 503-567 · Zbl 0555.54004 [11] van Mill, J., Infinite-dimensional topology, () · Zbl 1027.57022 [12] Oxtoby, J., Cartesian products of Baire spaces, Fund. math., 49, 157-166, (1961) · Zbl 0113.16402 [13] Pytkeev, E.G., The Baire property of spaces of continuous functions, Mat. zametki, 38, 726-740, (1985), (in Russian) · Zbl 0601.54032 [14] Tkachuk, V.V., Characterization of Baire property in Cp(X) by the properties of a space X, (), 21-27, (in Russian) [15] Dobrowolski, T.; Marciszewski, W.; Mogilski, J., Topological classification of function spaces of low Borel complexity, Trans. amer. math. soc., 328, 307-324, (1991) · Zbl 0768.54016
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.