Summary
A new class of locally convex linear topological spaces, the (db)-spaees, recently defined by Robertson, Tweddle and Yeomans, interpolates the classes of Baire-like and unordered Baire-like spaces. Saxon and Narayanaswami have given convex metrizable spaces that distinguish among these classes. This paper gives a new characterization of (db)-spaces from which is extracted the class of b-spaces. This class interpolates the classes of α-spaces and α′-spaces of Lehner. Let Cϰ(X) be the space of all real-valued continuous functions on the completely regular Sausdorf space X, supplied with the topology of uniform convergence on compact sets of X. It is shown that Cϰ(X) is a b-space if and only if it is an α-space. A characterization of X for which Cϰ(X) is a (db)-space is unknown. Other open questions are stated in the paper.
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References
H. Buchwalter,Parties bornées d'un espace topologique complètement régulier, Seminaire Choquet,9 (1969–1970), no. 14, pp. 15.
P. Dierolf -S. Dierolf -L. Drewnowski,Remarks and examples concerning unordered Baire-like and ultrabarrelled spaces, Colloq. Math.,39 (1978), pp. 109–116.
J. Horvàth,Topological Vector Spaces and Distributions, vol. I, Addison-Wesley, Reading, Mass. (1966).
J. L. Kelley,General Topology, D. Van Nostrand, Princeton, N.J. (1955).
W.Lehner,Über die Bedeutung gewisser Varianten des Baire'schen Kategorienbegriffs für die Funktionenraüme C c (T), Ludwig-Maximilians Universität Dissertation (1978).
L. Nachbin,Topological vector spaces of continuous functions, Proc. Nat. Acad. of Sci. U.S.A.,40 (1954), pp. 471–474.
W. J. Robertson -I. Tweddle -F. E. Yeomans,On the stability of barrelled topologies III, Bull. Australian Math. Society,22 (1980), pp. 99–112
W.Roelcke,Increasing sequences of absolutely convex sets, Cambridge talk 1970 cited inLehner [5].
S. A. Saxon,Nuclear and product space, Baire-like spaces and the strongest locally convex topology, Math. Ann.,197 (1972), pp. 87–206.
S. A. Saxon,Some normed barrelled spaces which are not Baire, Math. Ann.,209 (1974), pp. 153–160.
S. A. Saxon -P. P. Narayanaswami,Metrizable (LF)-spaces, (db)-spaces, and the separable quotient problem, Bull. Australian Math. Soc.,23 (1981), pp. 65–80.
J. Schmets,Espaces de Fonctions Continues, Lecture Notes in Math., n. 519, Springer-Verlag, Berlin, 1976.
T. Shirota,On locally convex vector spaces of continuous functions, Proc. Japan Acad.,30 (1954), pp. 294–298.
A. R. Todd -S. A. Saxson,A property of locally convex Baire spaces, Math. Ann.,206 (1973), pp. 23–34.
M. Valdivia,On certain barrelled normed spaces, Ann. Inst. Fourier (Grenoble),29 (1979), pp. 39–56.
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Todd, A.R. C ϰ(X) and a property of (db)-spaces. Annali di Matematica pura ed applicata 128, 317–323 (1981). https://doi.org/10.1007/BF01789479
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DOI: https://doi.org/10.1007/BF01789479