Abstract
Answering a long-standing open problem we characterize locally complete spaces \(C_{k}(X)\), where \(C_{k}(X)\) denotes the space of all real-valued continuous functions on a Tychonoff space X endowed with the compact-open topology. We show that \(C_{k}(X)\) is locally complete if and only if every strongly functionally compact-finite sequence of functionally closed subsets of X is locally finite. For the important partial case when X is a pseudocompact space, we extend a classical result of Warner with an independent proof.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Banakh, T., Gabriyelyan, S.: On the \(C_{k}\)-stable closure of the class of (separable) metrizable spaces. Monatshefte Math. 180, 39–64 (2016)
Banakh, T., Gabriyelyan, S.: Locally convex spaces with the strong Gelfand–Phillips property. Ann. Funct. Anal. 14:27, 30pp. (2023)
Buchwalter, H.: Fonctions continues et mesures sur un espace complètement régulier. Lecture Notes, Ecole d’été de Bruxelles 1972, 20 pages, à paraître en 1973
Buchwalter, H.: Sur le théorème de Nachbin-Shirota. J. Math. Pures Appl. 51, 399–415 (1972)
Buchwalter, H., Schmets, J.: Sur quelques propriétés de l’espace \(C_{s}(T)\). J. Math. Pures Appl., IX Sér. 52, 337–352 (1973)
Engelking, R.: General Topology. Heldermann Verlag, Berlin (1989)
Ferrando, J.C., Ka̧kol, J., Saxon, S.: Characterizing \(P\)-spaces \(X\) in terms of \(C_p(X)\). J. Convex Anal. 22(4), 905–915 (2015)
Frolík, Z.: The topological product of two pseudocompact spaces. Czech. Math. J. 10, 339–349 (1960)
Gabriyelyan, S.: On the Ascoli property for locally convex spaces. Topol. Appl. 230, 517–530 (2017)
Gabriyelyan, S.: Locally convex properties of free locally convex spaces. J. Math. Anal. Appl. 480, 123453 (2019)
Gabriyelyan, S.: Ascoli and sequentially Ascoli spaces. Topology Appl. 285(107401), 28 (2020)
Gabriyelyan, S.: Ascoli’s theorem for pseudocompact spaces. RACSAM 114(174), 10 (2020)
Haydon, R.: Compactness in \(C_{s}(T)\) and applications. Publ. Dep. Math. Lyon 9, 105–113 (1972)
Jarchow, H.: Locally Convex Spaces. B.G. Teubner, Stuttgart (1981)
Ka̧kol, J., Kubiś, W., Lopez-Pellicer, M.: Descriptive Topology in Selected Topics of Functional Analysis. Developments in Mathematics, Springer (2011)
Markov, A.A.: On free topological groups. Dokl. Akad. Nauk SSSR 31, 299–301 (1941)
Pérez Carreras, P., Bonet, J. Barrelled Locally Convex Spaces. North-Holland Mathematics Studies 131, North-Holland, Amsterdam (1987)
Warner, S.: The topology of compact convergence on continuous function spaces. Duke Math. J. 25, 265–282 (1958)
Wilansky, A.: Topology for Analysis. Ginn, New York (1970)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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
Gabriyelyan, S. Local completeness of \(C_{k}(X)\). Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 117, 152 (2023). https://doi.org/10.1007/s13398-023-01487-7
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s13398-023-01487-7