Remarks on a paper by Juh\'asz and Kunen

We give an equivalent, but simpler formulation of the axiom SEP introduced by Juhasz and Kunen. Our formulation shows that many of the consequences of the weak Freese-Nation Property of $\mathcal{P}(\omega )$ already follow from SEP. We show that it is consistent that SEP holds while $\mathcal{P}(\omega )$ fails to have the $({\mathrm{\aleph }}_1,{\mathrm{\aleph }}_0)$-ideal property, which has been introduced by Hart and Dow . This answers a question addressed independently by Fuchino and by Kunen. We also consider some natural variants of SEP and show that certain changes in the definition of SEP do not lead to a different principle. This answers a question addressed by Blass.