# Article

 Title: Sequential completeness of subspaces of products of two cardinals (English) Author: Frič, Roman Author: Kemoto, Nobuyuki Language: English Journal: Czechoslovak Mathematical Journal ISSN: 0011-4642 (print) ISSN: 1572-9141 (online) Volume: 49 Issue: 1 Year: 1999 Pages: 119-125 Summary lang: English . Category: math . Summary: Let $\kappa$ be a cardinal number with the usual order topology. We prove that all subspaces of $\kappa ^2$ are weakly sequentially complete and, as a corollary, all subspaces of $\omega _1^2$ are sequentially complete. Moreover we show that a subspace of $(\omega _1+1)^2$ need not be sequentially complete, but note that $X=A\times B$ is sequentially complete whenever $A$ and $B$ are subspaces of $\kappa$. (English) Keyword: sequentially continuous Keyword: sequentially complete Keyword: product space MSC: 54A20 MSC: 54B10 MSC: 54C08 idZBL: Zbl 0949.54004 idMR: MR1676829 . Date available: 2009-09-24T10:20:34Z Last updated: 2020-07-03 Stable URL: http://hdl.handle.net/10338.dmlcz/127472 . Reference: [F1] R. Frič: Sequential envelope and subspaces of the Čech-Stone compactification.In General Topology and its Relations to Modern Analysis and Algebra III (Proc. Third Prague Topological Sympos., 1971), Academia, Praha, 1971, pp. 123–126. MR 0353260 Reference: [F2] R. Frič: On E-sequentially regular spaces.Czechoslovak Math. J. 26 (1976), 604–612. MR 0428240 Reference: [FK] R. Frič and V. Koutník: Sequentially complete spaces.Czechoslovak Math. J. 29 (1979), 287–297. MR 0529516 Reference: [Ki] J. Kim: Sequentially complete spaces.J. Korean Math. Soc. 9 (1972), 39–43. Zbl 0242.54024, MR 0303508 Reference: [Ko] V. Koutník: On sequentially regular convergence spaces.Czechoslovak Math. J. 17 (1967), 232–247. MR 0215277 Reference: [KOT] N. Kemoto, H. Ohta and K. Tamano: Products of spaces of ordinal numbers.Top. Appl. 45 (1992), 245–260. MR 1180812 Reference: [No] J. Novák: On sequential envelope.In General Topology and its Relations to Modern Analysis and Algebra I (Proc. First Prague Topological Sympos., 1961 ), Publishing House of the Czecoslovak Academy of Sciences, Praha, 1962, pp. 292–294. MR 0175082 .

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