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Zbl 0772.54020
Bella, Angelo
More on the product of pseudo radial spaces.
(English)
[J] Commentat. Math. Univ. Carol. 32, No.1, 125-128 (1991). ISSN 0010-2628

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.\par 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 $\{x\sb \alpha$: $\alpha<\kappa\}$ converges to $p$ iff every neighborhood of $p$ contains a set of the form $\{x\sb \alpha$: $\beta\leq\alpha <\kappa\}$. 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 $\vert A\vert\geq \text{nw}(\text{cl }A)$ for every subset $A$ of $X$.\par 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 $A$.
[J.Roitman (Lawrence)]
MSC 2000:
*54D55 Sequential spaces
54A25 Cardinality properties of topological spaces
54B10 Product spaces (general topology)

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