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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 \(\kappa \).
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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