Pseudocompact spaces 𝑋 and 𝑑𝑓-spaces 𝐶_{𝑐}(𝑋)

Ka̧kol, Jerzy; Saxon, Stephen; Todd, Aaron

doi:10.1090/S0002-9939-04-07279-X

Pseudocompact spaces $X$ and $df$-spaces $C_{c}\left ( X\right )$
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by Jerzy Ka̧kol, Stephen A. Saxon and Aaron R. Todd PDF

Proc. Amer. Math. Soc. 132 (2004), 1703-1712 Request permission

Abstract:

Let $X$ be a completely regular Hausdorff space, and let $C_{c}\left ( X\right )$ be the space $C\left ( X\right )$ of continuous real-valued functions on $X$ endowed with the compact-open topology. We find various equivalent conditions for $C_{c}\left ( X\right )$ to be a $df$-space, resolving an old question of Jarchow and consolidating work by Jarchow, Mazon, McCoy and Todd. Included are analytic characterizations of pseudocompactness and an example that shows that, for $C_{c}\left ( X\right )$, Grothendieck’s $DF$ -spaces do not coincide with Jarchow’s $df$-spaces. Any such example necessarily answers a thirty-year-old question on weak barrelledness properties for $C_{c}\left ( X\right )$, our original motivation.

References

Henri Buchwalter and Jean Schmets, Sur quelques propriétés de l’espace $C_{s}$ $(T)$, J. Math. Pures Appl. (9) 52 (1973), 337–352 (French). MR 333687
D. H. Fremlin, D. J. H. Garling, and R. G. Haydon, Bounded measures on topological spaces, Proc. London Math. Soc. (3) 25 (1972), 115–136. MR 344405, DOI 10.1112/plms/s3-25.1.115
Hans Jarchow, Locally convex spaces, Mathematische Leitfäden. [Mathematical Textbooks], B. G. Teubner, Stuttgart, 1981. MR 632257
Jerzy Kąkol and Stephen A. Saxon, Montel (DF)-spaces, sequential (LM)-spaces and the strongest locally convex topology, J. London Math. Soc. (2) 66 (2002), no. 2, 388–406. MR 1920410, DOI 10.1112/S0024610702003459
J. Ka̧kol, S. A. Saxon, and A. R. Todd, Weak barrelledness for $C\left ( X\right )$ spaces, preprint.
W. Lehner, Über die Bedeutung gewisser Varianten des Baire’schen Kategorienbegriffs für die Funktionenräume $C_{c}\left ( T\right )$, Dissertation, Ludwig-Maximilians-Universität, München (1979).
José M. Mazón, On the df-spaces of continuous functions, Bull. Soc. Roy. Sci. Liège 53 (1984), no. 5, 225–232. MR 770088
R. A. McCoy and Aaron R. Todd, Support spaces and $P(K)$-points in the Stone-Čech compactification of the integers, Southeast Asian Bull. Math. 20 (1996), no. 1, 11–18. MR 1384520
Jan van Mill, Sixteen topological types in $\beta \omega -\omega$, Topology Appl. 13 (1982), no. 1, 43–57. MR 637426, DOI 10.1016/0166-8641(82)90006-2
Peter D. Morris and Daniel E. Wulbert, Functional representation of topological algebras, Pacific J. Math. 22 (1967), 323–337. MR 213876
Pedro Pérez Carreras and José Bonet, Barrelled locally convex spaces, North-Holland Mathematics Studies, vol. 131, North-Holland Publishing Co., Amsterdam, 1987. Notas de Matemática [Mathematical Notes], 113. MR 880207
Stephen A. Saxon, Nuclear and product spaces, Baire-like spaces, and the strongest locally convex topology, Math. Ann. 197 (1972), 87–106. MR 305010, DOI 10.1007/BF01419586
Stephen A. Saxon and L. M. Sánchez Ruiz, Dual local completeness, Proc. Amer. Math. Soc. 125 (1997), no. 4, 1063–1070. MR 1389535, DOI 10.1090/S0002-9939-97-03864-1
P. Erdös, On the distribution of normal point groups, Proc. Nat. Acad. Sci. U.S.A. 26 (1940), 294–297. MR 2000, DOI 10.1073/pnas.26.4.294
Seth Warner, The topology of compact convergence on continuous function spaces, Duke Math. J. 25 (1958), 265–282. MR 102735
Albert Wilansky, Modern methods in topological vector spaces, McGraw-Hill International Book Co., New York, 1978. MR 518316