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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:
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