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Further results on the product of chain-net spaces. (English) Zbl 0848.54004
A space is called a chain-net (or pseudoradial) space if for every nonclosed set \(F\) there exists a regular, infinite cardinal \(\kappa\) and a function \(f : \kappa \to F\) such that \(f \) converges to a point \(x \notin F\). The main result in the paper is the Theorem: If \(X\) is chain-net, Hausdorff, \([\lambda,\lambda]\)-compact with no convergent (nontrivial) sequence of length less than \(\lambda\), and \(Y\) is a \(\lambda\)-chain-net space (i.e., for every nonclosed set \(F\) there exists a function \(f : \lambda \to F\) such that \(f\) converges to a point \(x \notin F)\), then \(X \times Y\) is a chain-net space. From this the authors obtain a corollary (attributed to J. Gerlits and Z. Nagy) that says: the product of a chain-net, Hausdorff, countably compact space and a sequential space is again chain-net. The authors show that for chain-net spaces, the condition “there are no nontrivial convergent sequences of length less than \(\lambda\)” is equivalent to “\(\lambda\)-additive” (i.e., every intersection of less than \(\lambda\) open sets is open). The authors also discuss the (still open) question: is the product of two compact chain-net spaces a chain-net space? Z. Frolík and the third author [ibid. 5, 7-11 (1989; Zbl 0705.54020)] proved that the answer is “yes” if one of the spaces is Fréchet chain-net (i.e., radial).

54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
54D55 Sequential spaces
54B10 Product spaces in general topology