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More on the product of pseudo radial spaces. (English) Zbl 0772.54020
The main question is when the product of pseudo radial spaces is pseudo radial. The main theorem is that the product of a compact pseudo radial space and a compact monolithic pseudo radial space is pseudo radial.
Here are the definitions needed: Generalize the notion of a sequence to include \(\kappa\)-sequences, and convergence to include convergence by \(\kappa\)-sequences, where the sequence \(\{x_ \alpha\): \(\alpha<\kappa\}\) converges to \(p\) iff every neighborhood of \(p\) contains a set of the form \(\{x_ \alpha\): \(\beta\leq\alpha <\kappa\}\). A pseudo radial space is one in which every subset \(A\) which is not closed contains a sequence which converges outside of \(A\). A space \(X\) is monolithic iff \(| A|\geq \text{nw}(\text{cl }A)\) for every subset \(A\) of \(X\).
The proof also uses the notion of the chain character of a pseudo radial space, i.e. the smallest \(\kappa\) so that if \(A\) is not closed then there is a \(\leq\kappa\)-sequence in \(A\) converging to some point not in \(A\).

MSC:
54D55 Sequential spaces
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
54B10 Product spaces in general topology
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