Abstract
A compact space X is called \(\pi\)-monolithic if for any surjective continuous mapping \(f \colon X \rightarrow K\) where K is a metrizable compact space there exists a metrizable compact space \(T \subseteq X\) such that \(f(T)=K\). A topological space X is Baire if the intersection of any sequence of open dense subsets of X is dense in X. Let \(C_p(X, Y)\) denote the space of all continuous Y-valued functions C(X,Y) on a Tychonoff space X with the topology of pointwise convergence. In this paper we have proved that for a totally disconnected space X the space \(C_p(X,\{0,1\})\) is Baire if, and only if, \(C_p(X,K)\) is Baire for every \(\pi\)-monolithic compact space K.
For a Tychonoff space X the space \(C_p(X,\mathbb{R})\) is Baire if, and only if, \(C_p(X,L)\) is Baire for each Fréchet space L.
We construct a totally disconnected Tychonoff space T such that \(C_p(T,M)\) is Baire for a separable metric space M if, and only if, M is a Peano continuum. Moreover, \(C_p(T,[0,1])\) is Baire but \(C_p(T,\{0,1\})\) is not.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. V. Arhangel’skii, Topological Function Spaces, MoscowState University (1989).
A. V. Arhangel’skii and V. I. Ponomarev, Fundamentals of General Topology: Problems and Exercises, Springer (2001).
R. Engelking, General Topology, Revised and completed edition, Heldermann Verlag (Berlin, 1989).
D. J. Lutzer and R. A. McCoy, Category in function spaces. I, Pacific J. Math., 90 (1980), 145–168.
Eric K. van Douwen, Collected Papers, vol. 1 (J. van Mill, Ed.), North-Holland (Amsterdam, 1994).
A. V. Osipov, Baire property of space of Baire-one functions, arXiv:2110.15496 (2021).
A. V. Osipov and E. G. Pytkeev, Baire property of spaces of [0, 1]-valued continuous functions, arXiv:2203.05976 (2022).
E. G. Pytkeev, Baire property of spaces of continuous functions, Math. Notes, 38 (1985), 908–915; translated from Mat. Zametki, 38 (1985), 726–740).
H. H. Schaefer, Topological Vector Spaces, Springer-Verlag (New York, 1971).
D. B. Shakhmatov, A pseudocompact Tychonoff space all countable subsets of which are closed and C*-embedded, Topology Appl., 22 (1986), 139–144.
V. V. Tkachuk, Characterization of the Baire property in Cp(X) by the properties of the space X, Research papers, Topology – Maps and Extensions of Topological Spaces (Ustinov) (1985), 21–27.
V. V. Tkachuk, A Cp-theory Problem Book. Topological and Function Spaces, Springer (2011).
H. Toruńczyk, Characterizing Hilbert space topology, Fund. Math., 111 (1981), 247– 262.
G. Vidossich, On topological spaces whose functions space is of second category, Invent. Math., 8 (1969), 111–113.
Acknowledgement
The authors are grateful to the referee for useful remarks and suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
The research funding from the Ministry of Science and Higher Education of the Russian Federation (Ural Federal University Program of Development within the Priority-2030 Program) is gratefully acknowledged.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Osipov, A.V., Pytkeev, E.G. Baire property of some function spaces. Acta Math. Hungar. 168, 246–259 (2022). https://doi.org/10.1007/s10474-022-01274-7
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10474-022-01274-7
Key words and phrases
- function space
- Baire property
- \(\pi\)-monolithic compact space
- Fréchet space
- totally disconnected space
- Peano continuum