More on the product of pseudo radial spaces.

*(English)*Zbl 0772.54020The 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 \(A\).

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 \(A\).

Reviewer: J.Roitman (Lawrence)