Products of $\mathfrak {m}$-compact spaces
HTML articles powered by AMS MathViewer
- by Victor Saks and R. M. Stephenson
- Proc. Amer. Math. Soc. 28 (1971), 279-288
- DOI: https://doi.org/10.1090/S0002-9939-1971-0273570-6
- PDF | Request permission
Abstract:
Some results are given on the closure under suitably restricted products of a class of spaces similar to one considered by Z. Frolík and, more recently, by N. Noble. An answer is given to the following question of Gulden, Fleischman, and Weston: Does there exist $\mathfrak {M} > {\aleph _0}$ and an $\mathfrak {M}$-compact space $X$ such that some subset $A$ of $X$ of cardinality $\leqq \mathfrak {M}$ is contained in no compact subset of $X$? It is shown that for every $\mathfrak {M} \geqq {\aleph _0}$ there is a topological group which has this property.References
- Zdeněk Frolík, Generalisations of compact and Lindelöf spaces, Czechoslovak Math. J. 9(84) (1959), 172–217 (Russian, with English summary). MR 105075, DOI 10.21136/CMJ.1959.100348
- Zdeněk Frolík, The topological product of countably compact spaces, Czechoslovak Math. J. 10(85) (1960), 329–338 (English, with Russian summary). MR 117705, DOI 10.21136/CMJ.1960.100417
- Zdeněk Frolík, The topological product of two pseudocompact spaces, Czechoslovak Math. J. 10(85) (1960), 339–349 (English, with Russian summary). MR 116304, DOI 10.21136/CMJ.1960.100418
- 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
- Irving Glicksberg, Stone-Čech compactifications of products, Trans. Amer. Math. Soc. 90 (1959), 369–382. MR 105667, DOI 10.1090/S0002-9947-1959-0105667-4
- S. L. Gulden, W. M. Fleischman, and J. H. Weston, Linearly ordered topological spaces, Proc. Amer. Math. Soc. 24 (1970), 197–203. MR 250272, DOI 10.1090/S0002-9939-1970-0250272-2 T. Nakayama, Sets, topology, and algebraic systems, Shibundo, Tokyo, 1949, p. 138.
- S. Negrepontis, An example on realcompactifications, Arch. Math. (Basel) 20 (1969), 162–164. MR 244952, DOI 10.1007/BF01899007
- Norman Noble, Countably compact and pseudo-compact products, Czechoslovak Math. J. 19(94) (1969), 390–397. MR 248717, DOI 10.21136/CMJ.1969.100911
- J. Novák, On the Cartesian product of two compact spaces, Fund. Math. 40 (1953), 106–112. MR 60212, DOI 10.4064/fm-40-1-106-112
- Mary Ellen Rudin, A technique for constructing examples, Proc. Amer. Math. Soc. 16 (1965), 1320–1323. MR 188976, DOI 10.1090/S0002-9939-1965-0188976-0
- C. T. Scarborough and A. H. Stone, Products of nearly compact spaces, Trans. Amer. Math. Soc. 124 (1966), 131–147. MR 203679, DOI 10.1090/S0002-9947-1966-0203679-7
- J. E. Vaughan, Spaces of countable and point-countable type, Trans. Amer. Math. Soc. 151 (1970), 341–351. MR 266157, DOI 10.1090/S0002-9947-1970-0266157-6
- R. Grant Woods, Some $\aleph _{O}$-bounded subsets of Stone-Čech compactifications, Israel J. Math. 9 (1971), 250–256. MR 278266, DOI 10.1007/BF02771590
Bibliographic Information
- © Copyright 1971 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 28 (1971), 279-288
- MSC: Primary 54.52
- DOI: https://doi.org/10.1090/S0002-9939-1971-0273570-6
- MathSciNet review: 0273570