On Psuedo-Radial Spaces

If a space is compact then sequential compactness is equivalent to the statement 'every closed set is generated by converging (countable) sequences'. A subset of a pseudo-radial space is closed iff it can be generated by convergent sequences (dropping the restriction that the sequence has to be countable). Hence, pseudo-radiality is a generalization of sequential compactness module compactness.  

It is known that if $\mathfrak{c} \leq \aleph_2$ then compactness plus sequential compactness (CSC for short) implies pseudo-radiality. On the other hand, if $\mathfrak{c} = \aleph_3$ there is a CSC space which fails to be pseudo-radial.

This is a panoramic talk. The techniques inspired by Sapirovskii are worth it to review. Also, we present other known important results that involve the splitting number $\mathfrak{s}$.