A theorem on the cardinality of $\kappa$-total spaces
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- by R. M. Stephenson PDF
- Proc. Amer. Math. Soc. 89 (1983), 367-370 Request permission
Abstract:
Throughout this article, $\kappa$ denotes an arbitrary infinite cardinal number. In 1979, A. A. Gryzlov strengthened a well-known result of A. V. Arhangel’skii by proving that every compact ${T_1}$-space of pseudocharacter $\kappa$ has cardinality $\leqslant {2^\kappa }$. Using techniques similar to Gryzlov’s, we prove that every ${2^\kappa }$-total, ${T_1}$-space of pseudocharacter $\leqslant \kappa$ is compact and hence of cardinality $\leqslant {2^\kappa }$. Some related results and examples are given.References
-
P. Alexandroff and P. Urysohn, Mémoire sur les éspaces topologiques compacts, Ver. Akad. Wetensch. Amsterdam 14 (1929), 1-96.
- C. E. Aull, A certain class of topological spaces, Prace Mat. 11 (1967), 49–53. MR 0227914
- Alan Dow, Absolute $C$-embedding of spaces with countable character and pseudocharacter conditions, Canadian J. Math. 32 (1980), no. 4, 945–956. MR 590657, DOI 10.4153/CJM-1980-072-3
- 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
- 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
- A. A. Gryzlov, Two theorems on the cardinality of topological spaces, Dokl. Akad. Nauk SSSR 251 (1980), no. 4, 780–783 (Russian). MR 568530
- Victor Saks, Ultrafilter invariants in topological spaces, Trans. Amer. Math. Soc. 241 (1978), 79–97. MR 492291, DOI 10.1090/S0002-9947-1978-0492291-9
- Victor Saks and R. M. Stephenson Jr., Products of ${\mathfrak {m}}$-compact spaces, Proc. Amer. Math. Soc. 28 (1971), 279–288. MR 273570, DOI 10.1090/S0002-9939-1971-0273570-6
- J. E. Vaughan, Powers of spaces of nonstationary ultrafilters, Fund. Math. 117 (1983), no. 1, 5–14. MR 712208, DOI 10.4064/fm-117-1-5-14
Additional Information
- © Copyright 1983 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 89 (1983), 367-370
- MSC: Primary 54A25; Secondary 54D10, 54G20
- DOI: https://doi.org/10.1090/S0002-9939-1983-0712653-9
- MathSciNet review: 712653