×

The metrizability and completeness of the \(\sigma \)-compact-open topology on \(C^*(X)\). (English) Zbl 1256.54046

Let \(X\) be a Tychonoff space. On the set \(C ( X )\) of all real-valued continuous functions on \(X\) and on the set \(C^* ( X )\) of all bounded real-valued continuous functions on \(X\) quite a few natural topologies are defined. Among them are the classical compact-open topology \(K\) and the topology of uniform convergence \(u\).
Another natural locally convex topology \(C^* ( X )\) between \(k\) and \( u\) is the \(\sigma\)-compact-open topology \(\sigma\). The space \(C^* ( X )\) with the topology \(\sigma\) is denoted by \(C^*_\sigma (X)\). This topology, which can also be considered as the topology uniform convergence on \(\sigma\)-compact subsets of \(X\), was introduced and studied by D. Gulick in the paper [Math. Scand. 30, 159-176 (1972; Zbl 0253.46045)], where the metrizability and uniform completeness of \(C^*_\sigma (X)\) were studied briefly.
In this paper the authors continue the study of the space \(C^*_\sigma (X)\) obtaining further results in the context of metrizability and completeness. More precisely, they prove a number of properties of \(C^*_\sigma (X)\) equivalent to its metrizability (Theorems 2.1 and 2.4). Also various kinds of completeness of \(C^*_\sigma (X)\) are studied such as,
\(\bullet\) Uniform completeness of \(C^*_\sigma (X)\) (Theorem 3.2).
\(\bullet\) Čech-completeness, local Čech-completeness, Sieve-completeness and partition-completeness of \(C^*_\sigma (X)\) ( Theoren 3.5)
\(\bullet\) Almost Čech-completeness and pseudo-completeness of \(C^*_\sigma (X)\) ( Theorem 3.6).
Since \(C^*_\sigma (X)\) is a locally convex space, the authors examine the case when this space is barreled or bornological. Also they study the conditions under which the space \(C^*_\sigma (X)\) can be a nuclear or a Schwartz space ( Theorem 4.1).
This is an interesting paper which continues and improves the previous work of the first author.

MSC:

54C35 Function spaces in general topology
54E18 \(p\)-spaces, \(M\)-spaces, \(\sigma\)-spaces, etc.
54E35 Metric spaces, metrizability
54E50 Complete metric spaces
54E52 Baire category, Baire spaces
54E99 Topological spaces with richer structures
46A08 Barrelled spaces, bornological spaces
46A25 Reflexivity and semi-reflexivity
46A99 Topological linear spaces and related structures
46E10 Topological linear spaces of continuous, differentiable or analytic functions

Citations:

Zbl 0253.46045
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Aarts, J. M.; Lutzer, D. J., Pseudo-completeness and the product of Baire spaces, Pacific J. Math., 48, 1-10 (1973) · Zbl 0238.54027
[2] Anderson, R. D., Hilbert space is homeomorphic to the countable infinite product of lines, Bull. Amer. Math. Soc., 72, 515-519 (1966) · Zbl 0137.09703
[3] Arens, R., A topology for spaces of transformations, Ann. of Math., 47, 480-495 (1946) · Zbl 0060.39704
[4] Arens, R.; Dugundji, J., Topologies for function spaces, Pacific J. Math., 1, 5-31 (1951) · Zbl 0044.11801
[5] Arhangelʼskiĭ, A. V., On linear homeomorphisms of function spaces, Soviet Math. Dokl., 25, 852-855 (1982) · Zbl 0522.54015
[6] Arhangelʼskiĭ, A. V.; Tkachenko, M., Topological Groups and Related Structures, Atlantis Stud. Math., vol. 1 (2008), Atlantis Press: Atlantis Press Amsterdam, Paris
[7] C.E. Aull, Some base axioms for topology involving enumerability, in: General Topology and Its Relations to Modern Analysis and Algebra, Proceedings of the Kanpur Topological Conference, 1968, pp. 55-61.; C.E. Aull, Some base axioms for topology involving enumerability, in: General Topology and Its Relations to Modern Analysis and Algebra, Proceedings of the Kanpur Topological Conference, 1968, pp. 55-61.
[8] Buck, R. C., Bounded continuous functions on a locally compact space, Michigan Math. J., 5, 95-104 (1958) · Zbl 0087.31502
[9] Buhagiar, D.; Yoshioka, I., Sieves and completeness properties, Questions Answers Gen. Topology, 18, 143-162 (2000) · Zbl 0970.54024
[10] Chaber, J.; Čoban, M. M.; Nagami, K., On monotonic generalizations of Moore spaces, Čech-complete spaces and \(p\)-spaces, Fund. Math., 84, 2, 107-119 (1974) · Zbl 0292.54038
[11] Cristescu, Romulus, Topological Vector Spaces (1977), Noordhoff International Publishing: Noordhoff International Publishing Leyden, The Netherlands, (English translation) · Zbl 0345.46001
[12] Engelking, R., General Topology (1989), Heldermann Verlag: Heldermann Verlag Berlin · Zbl 0684.54001
[13] Fox, R. H., On topologies for function spaces, Bull. Amer. Math. Soc., 51, 429-432 (1945) · Zbl 0060.41202
[14] Frolík, Z., Generalizations of the \(G_\delta \)-property of complete metric spaces, Czechoslovak Math. J., 10, 359-379 (1960) · Zbl 0100.18701
[15] Giles, R., A generalization of the strict topology, Trans. Amer. Math. Soc., 161, 467-474 (1971) · Zbl 0204.43603
[16] Gillman, L.; Jerison, M., Rings of Continuous Functions (1960), D. Van Nostrand: D. Van Nostrand Princeton, NJ · Zbl 0093.30001
[17] Gruenhage, G., Generalized metric spaces, (Kunen, K.; Vaughan, J. E., Handbook of Set-Theoretic Topology (1984), Elsevier Science Publishers B. V.), 423-501
[18] Gulick, D., The \(σ\)-compact-open topology and its relatives, Math. Scand., 30, 159-176 (1972) · Zbl 0253.46045
[19] Jackson, J. R., Comparison of topologies on function spaces, Proc. Amer. Math. Soc., 3, 156-158 (1952) · Zbl 0046.11703
[20] Jarchow, Hans, Locally Convex Spaces (1981), B.G. Teubner: B.G. Teubner Stuttgart · Zbl 0466.46001
[21] Kundu, S., \(C_b(X)\) revisited: induced map and submetrizability, Quaest. Math., 27, 47-56 (2004) · Zbl 1064.54031
[22] Kundu, S.; McCoy, R. A., Topologies between compact and uniform convergence on function spaces, Internat. J. Math. Math. Sci., 16, 101-109 (1993) · Zbl 0798.54019
[23] Kundu, S.; McCoy, R. A.; Raha, A. B., Topologies between compact and uniform convergence on function spaces. II, Real Anal. Exchange, 18, 176-189 (1992/1993) · Zbl 0798.54020
[24] Michael, E., Complete spaces and tri-quotient maps, Illinois J. Math., 21, 3, 716-733 (1977) · Zbl 0386.54007
[25] Michael, E., A note on completely metrizable spaces, Proc. Amer. Math. Soc., 96, 3, 513-522 (1986) · Zbl 0593.54028
[26] Michael, E., Almost complete spaces, hypercomplete spaces and related mapping theorems, Topology Appl., 41, 1-2, 113-130 (1991) · Zbl 0749.54003
[27] Michael, E., Partition-complete spaces are preserved by tri-quotient maps, Topology Appl., 44, 235-240 (1992) · Zbl 0794.54018
[28] Munkres, James R., Topology (2000), Prentice-Hall Inc.: Prentice-Hall Inc. New Jersey · Zbl 0951.54001
[29] Narici, L.; Beckenstein, E., Topological Vector Spaces (1985), Marcel Dekker: Marcel Dekker New York · Zbl 0569.46001
[30] Oxtoby, J. C., Cartesian products of Baire spaces, Fund. Math., 49, 157-166 (1961) · Zbl 0113.16402
[31] Schemets, J., Espaces de fonctions continues, Lecture Notes in Math., vol. 519 (1976), Springer-Verlag: Springer-Verlag New York
[32] Sentilles, F. D., Bounded continuous functions on a completely regular space, Trans. Amer. Math. Soc., 168, 311-336 (1972) · Zbl 0244.46027
[33] van Douwen, Eric K., An unBaireable stratifiable space, Proc. Amer. Math. Soc., 67, 2, 324-326 (1977) · Zbl 0379.54008
[34] Wheeler, Robert F., A survey of Baire measures and strict topologies, Expo. Math., 2, 97-190 (1983) · Zbl 0522.28009
[35] Wicke, H. H., Open continuous images of certain kinds of \(M\)-spaces and completeness of mappings and spaces, Gen. Topology Appl., 1, 1, 85-100 (1971) · Zbl 0212.27203
[36] Wicke, H. H.; Worrell, J. M., On the open continuous images of paracompact Čech-complete spaces, Pacific J. Math., 37, 265-275 (1971) · Zbl 0197.19202
[37] Willard, Stephen, General Topology (1970), Addison-Wesley Publishing Co.: Addison-Wesley Publishing Co. Reading, MA, London, Don Mills, ON · Zbl 1052.54001
[38] Yosida, Kosaku, Functional Analysis (1980), Springer-Verlag: Springer-Verlag Heidelberg · Zbl 0435.46002
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.