In this paper we study the Baire property and the Choquet property for the space $K_1(X,Y)$ – first functional class Lebesgue mappings, where $X$ is a Tychonoff space and $Y\in \{\mathbb{R}, [0,1], \{0,1\}\}$. It is proved that the space $B_1(X,[0,1])$ – $[0,1]$-valued Baire mappings of the first class is a Choquet (Baire) space if and only if the space $K_1(X,\{0,1\})$ is a Choquet (Baire) space. The obtained studies allow us to quite simply solve V. Tkachuk's question about the coincidence of pseudocompactness and pseudocompleteness of the space $C_p(X,[0,1])$.
Keywords: Baire property, Choquet space, Baire functions, Lebesgue functional class mappings, function space
Received April 25, 2025
Revised June 24, 2025
Accepted June 30, 2025
Alexander Vladimirovich Osipov, Dr. Phys.-Math. Sci., Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia; Ural Federal University, Yekaterinburg, 620083 Russia, e-mail: OAB@list.ru
REFERENCES
1. Aleksandrov P.S., Pasynkov B.A. Vvedeniye v teoriyu razmernosti [Introduction to the dimensionality theory]. Moscow, Mir Publ., 1973, 575 p.
2. Oxtoby J.C. Measure and category. A survey of the analogies between topological and measure spaces. New York, Springer, 1971. doi: 10.1007/978-1-4615-9964-7. Translated to Russian under the title Mera i kategoriya, Moscow, Mir Publ, 1974, 158 p.
3. Lutzer D., McCoy R. Category in function spaces. I. Pacific J. Math., 1980, vol. 90, no. 1, pp. 145–168. https://doi.org/10.2140/pjm.1980.90.145
4. Pytkeev E.G. Baire property of spaces of continuous functions. Math. Notes, 1985, vol. 38, no. 5, pp. 908–915. https://doi.org/10.1007/BF01157538
5. Tkachuk V. Characterization of the Baire property in $C_p(X)$ by the properties of the space X. Research papers in Topology–Maps and extensions of topological spaces (Ustinov), 1985, pp. 21–27.
6. Douwen E.K. van Collected papers. Vol. 1 / ed. J. van Mill. Amsterdam: North-Holland Publ. Co., 1994. 767 p. ISBN-10: 0444816259 .
7. Banakh T., Gabriyelyan S. Baire category properties of some Baire type function spaces. Topol. Appl., 2020, vol. 272, art. no. 107078. https://doi.org/10.1016/j.topol.2020.107078
8. Osipov A.V. Baire property of space of Baire-one functions // Eur. J. Math. 2025. Vol. 11. Art. no. 8. 25 p. https://doi.org/10.1007/s40879-024-00799-1
9. Lukeš J., Malý J., Zajíček L. Fine topology methods in real analysis and potential theory. Lecture Notes in Mathematics Ser., vol. 1189. Berlin, Heidelberg, Springer, 1986, p. 476. https://doi.org/10.1007/BFb0075894
10. Sakai M. Two properties of $C_p(X)$ weaker than the Fréchet — Urysohn property. Topol. Appl., 2006, vol. 153, no. 15, pp. 2795–2804. https://doi.org/10.1016/j.topol.2005.11.012
11. Kąkol J., Kurka O., Leiderman A. Some classes of topological spaces extending the class of Δ-spaces. Proc. Amer. Math. Soc., 2024, vol. 152, no. 2, pp. 883–898. https://doi.org/10.1090/proc/16661
12. Osipov A.V. Baireness of the space of pointwise stabilizing functions of the first Baire class. Topol. Appl., 2025, vol. 362, art. no. 109218, 5 p. https://doi.org/10.1016/j.topol.2025.109218
13. Karlova O., Mykhaylyuk V. On stable Baire classes. Acta Math. Hungar., 2016, vol. 150, no. 1, pp. 36–48. https://doi.org/10.1007/s10474-016-0636-8
14. Osipov A.V., Pytkeev E.G. Baire property of spaces of [0,1]-valued continuous functions. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat, 2023, vol. 117, no. 1, art. no. 38, 10 p.
https://doi.org/10.1007/s13398-022-01371-w
15. Osipov A.V. On the Baire property of the space of Baire indicator functions. Mat. Zam., 2025, vol. 118, no. 4, pp. 564–574 (in Russian).
16. Kuratowski K. Topology, vol. 1. NY, Acad. Press, 1966, 560 p. https://doi.org/10.2307/3611898. Translated to Russian under the title Topologiya, Moscow, Mir Publ., 1966, 594 p.
17. Tkachuk V.V. A $C_p$-theory problem book. Topological and function spaces. NY, Springer, 2011, 488 p. https://doi.org/10.1007/978-1-4419-7442-6
18. Oxtoby J.C. The Banach–Mazur game and Banach category theorem. In: Contributions to the theory of games, Vol. 3; Ann. of Math. Stud., no. 39, NJ, Princeton, Princeton Univer. Press, 1957. P. 159–163.
19. Osipov A.V. The $\Delta_1$-property of X is equivalent to the Choquet property of $B_1(X)$. Topol. Appl., 2025, vol. 370, no. art. 109395, 6 p. https://doi.org/10.1016/j.topol.2025.109395
20. Banakh T., Hryniv O. Some Baire category properties of topological groups. Visnyk Lviv. Univ. Ser. Mech.-Mat., 2018, vol. 86, pp. 71–76. https://doi.org/10.48550/arXiv.1901.01420
21. Karlova O. On α-embedded sets and extension of mappings. Comment. Math. Univ. Carolin., 2013, vol. 54, no. 3, pp. 377–396. https://doi.org/10.48550/arXiv.1407.6155
22. Hrušák M., Tamariz-Mascarúa Á., Tkachenko M. Pseudocompact topological spaces. A survey of classic and new results with open problems. Developments in Mathematics Ser., vol. 55. Cham, Springer, 2018, 299 p. https://doi.org/10.1007/978-3-319-91680-4
Cite this article as: A.V. Osipov. On the properties of completeness type of spaces of first functional class Lebesgue mappings. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2025, vol. 31, no. 3, pp. 200–214.
