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Sequential completeness of subspaces of products of two cardinals. (English) Zbl 0949.54004
Summary: Let $$\kappa$$ be a cardinal number with the usual order topology. It is proved that all subspaces of $$\kappa ^2$$ are weakly sequentially complete and, as a corollary, all subspaces of $$\omega ^2_1$$ are sequentially complete. Moreover, it is shown 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 $$\kappa$$.
##### MSC:
 54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.) 54B10 Product spaces in general topology 54C08 Weak and generalized continuity
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##### References:
 [1] 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. [2] R. Frič: On E-sequentially regular spaces. Czechoslovak Math. J. 26 (1976), 604-612. · Zbl 0339.54005 [3] R. Frič and V. Koutník: Sequentially complete spaces. Czechoslovak Math. J. 29 (1979), 287-297. · Zbl 0401.54020 [4] J. Kim: Sequentially complete spaces. J. Korean Math. Soc. 9 (1972), 39-43. · Zbl 0242.54024 [5] V. Koutník: On sequentially regular convergence spaces. Czechoslovak Math. J. 17 (1967), 232-247. · Zbl 0173.24903 [6] N. Kemoto, H. Ohta and K. Tamano: Products of spaces of ordinal numbers. Top. Appl. 45 (1992), 245-260. · Zbl 0789.54006 [7] 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. · Zbl 0139.16001
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