# Article

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Keywords:
sequentially continuous; sequentially complete; product space
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 $\kappa$.
References:
[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
[F2] R. Frič: On E-sequentially regular spaces. Czechoslovak Math. J. 26 (1976), 604–612. MR 0428240
[FK] R. Frič and V. Koutník: Sequentially complete spaces. Czechoslovak Math. J. 29 (1979), 287–297. MR 0529516
[Ki] J. Kim: Sequentially complete spaces. J. Korean Math. Soc. 9 (1972), 39–43. MR 0303508 | Zbl 0242.54024
[Ko] V. Koutník: On sequentially regular convergence spaces. Czechoslovak Math. J. 17 (1967), 232–247. MR 0215277
[KOT] N. Kemoto, H. Ohta and K. Tamano: Products of spaces of ordinal numbers. Top. Appl. 45 (1992), 245–260. MR 1180812
[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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