On $\mu$-spaces and $k_{R}$-spaces
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- by J. L. Blasco PDF
- Proc. Amer. Math. Soc. 67 (1977), 179-186 Request permission
Abstract:
In this paper it is proved that when X is a ${k_R}$-space then $\mu X$ (the smallest subspace of $\beta X$ containing X with the property that each of its bounded closed subsets is compact) also is a ${k_R}$-space; an example is given of a ${k_R}$-space X such that its Hewitt realcompactification, $\upsilon X$, is not a ${k_R}$-space. We show with an example that there is a non-${k_R}$-space X such that $\upsilon X$ and $\mu X$ are ${k_R}$-spaces. Also we answer negatively a question posed by Buchwalter: Is $\mu X$ the union of the closures in $\upsilon X$ of the bounded subsets of X? Finally, without using the continuum hypothesis, we give an example of a locally compact space X of cardinality ${\aleph _1}$ such that $\upsilon X$ is not a k-space.References
- Nicolas Bourbaki, Sur certains espaces vectoriels topologiques, Ann. Inst. Fourier (Grenoble) 2 (1950), 5â16 (1951) (French). MR 42609, DOI 10.5802/aif.16
- Henri Buchwalter, Parties bornĂ©es dâun espace topologique complĂštement rĂ©gulier, SĂ©minaire Choquet: 1969/70, Initiation Ă lâAnalyse, SecrĂ©tariat mathĂ©matique, Paris, 1970, pp. Fasc. 2, Exp. 14, 15 (French). MR 0286066
- W. W. Comfort, On the Hewitt realcompactification of a product space, Trans. Amer. Math. Soc. 131 (1968), 107â118. MR 222846, DOI 10.1090/S0002-9947-1968-0222846-1
- J. A. DieudonnĂ©, Recent developments in the theory of locally convex vector spaces, Bull. Amer. Math. Soc. 59 (1953), 495â512. MR 62334, DOI 10.1090/S0002-9904-1953-09752-X
- Leonard Gillman and Meyer Jerison, Rings of continuous functions, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0116199, DOI 10.1007/978-1-4615-7819-2
- J. F. Kennison, Reflective functors in general topology and elsewhere, Trans. Amer. Math. Soc. 118 (1965), 303â315. MR 174611, DOI 10.1090/S0002-9947-1965-0174611-9 T. KĆmura and Y. KĆmura, Sur les espaces parfaits de suites et leurs gĂ©nĂ©ralizations, J. Math. Soc. Japan 15 (1963), 317-338. MR 28 #451.
- Gottfried Köthe, Topological vector spaces. I, Die Grundlehren der mathematischen Wissenschaften, Band 159, Springer-Verlag New York, Inc., New York, 1969. Translated from the German by D. J. H. Garling. MR 0248498
- Manuel LĂłpez Pellicer, On some not completely regular topological spaces, Rev. Mat. Hisp.-Amer. (4) 36 (1976), no. 4, 125â132 (Spanish, with English summary). MR 438285
- Saunders MacLane, Categories for the working mathematician, Graduate Texts in Mathematics, Vol. 5, Springer-Verlag, New York-Berlin, 1971. MR 0354798
- Ernest Michael, Local compactness and Cartesian products of quotient maps and $k$-spaces, Ann. Inst. Fourier (Grenoble) 18 (1968), no. fasc. 2, 281â286 vii (1969) (English, with French summary). MR 244943, DOI 10.5802/aif.300
- Leopoldo Nachbin, Topological vector spaces of continuous functions, Proc. Nat. Acad. Sci. U.S.A. 40 (1954), 471â474. MR 63647, DOI 10.1073/pnas.40.6.471
- S. Negrepontis, An example on realcompactifications, Arch. Math. (Basel) 20 (1969), 162â164. MR 244952, DOI 10.1007/BF01899007 N. Noble, Doctoral dissertation, Univ. of Rochester, Rochester, N. Y., 1967.
- Vlastimil PtĂĄk, On complete topological linear spaces, Äehoslovack. Mat. Ćœ. 3(78) (1953), 301â364 (Russian, with English summary). MR 0064303
- Taira Shirota, On locally convex vector spaces of continuous functions, Proc. Japan Acad. 30 (1954), 294â298. MR 64389
- Manuel Valdivia, On certain topologies on a vector space, Manuscripta Math. 14 (1974), 241â247. MR 361706, DOI 10.1007/BF01171410
- Seth Warner, The topology of compact convergence on continuous function spaces, Duke Math. J. 25 (1958), 265â282. MR 102735
- M. De Wilde and J. Schmets, CaractĂ©risation des espaces $C(X)$ ultrabornologiques, Bull. Soc. Roy. Sci. LiĂšge 40 (1971), 119â121 (French, with English summary). MR 291781
Additional Information
- © Copyright 1977 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 67 (1977), 179-186
- MSC: Primary 54D15
- DOI: https://doi.org/10.1090/S0002-9939-1977-0464152-7
- MathSciNet review: 0464152