On a condition for the pseudo radiality of a product.

*(English)*Zbl 0773.54006The 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 semi 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 \(\{x_ \alpha: \beta\leq \alpha< \kappa\}\). The convergence is strict if, for every \(\beta<\kappa\), \(x\not\in\text{cl}\{x_ \alpha: \alpha< \beta\}\). A pseudo radial space is one in which every set \(A\) which is not closed contains a sequence which converges outside of \(A\). A set \(A\) is \(\kappa\)-closed iff \(A\) contains \(\text{cl }B\), for all \(B\in[A]^{\leq\kappa}\). A space \(X\) is semi radial iff for every \(\kappa\) and every non-\(\kappa\)-closed set \(A\) there is a \(\lambda\leq\kappa\) and a \(\lambda\)-sequence in \(A\) which converges outside of \(A\).

Pseudo radiality is a well-known property. Semi radiality is defined in this paper. Semi radial implies pseudo radial. They also consider another property, defined from the chain character, and show that the product of two compact spaces, one of which is pseudo radial and the other of which has this property, is again 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 \(\{x_ \alpha: \beta\leq \alpha< \kappa\}\). The convergence is strict if, for every \(\beta<\kappa\), \(x\not\in\text{cl}\{x_ \alpha: \alpha< \beta\}\). A pseudo radial space is one in which every set \(A\) which is not closed contains a sequence which converges outside of \(A\). A set \(A\) is \(\kappa\)-closed iff \(A\) contains \(\text{cl }B\), for all \(B\in[A]^{\leq\kappa}\). A space \(X\) is semi radial iff for every \(\kappa\) and every non-\(\kappa\)-closed set \(A\) there is a \(\lambda\leq\kappa\) and a \(\lambda\)-sequence in \(A\) which converges outside of \(A\).

Pseudo radiality is a well-known property. Semi radiality is defined in this paper. Semi radial implies pseudo radial. They also consider another property, defined from the chain character, and show that the product of two compact spaces, one of which is pseudo radial and the other of which has this property, is again pseudo radial.

Reviewer: J.Roitman (Lawrence)