Reflection and weakly collectionwise Hausdorff spaces

Abstract

Abstract. We show that 0(0) implies that there is a first countable < 6-collectionwise Hausdorff space that is not weakly (9-collectionwise Hausdorff. We also show that in the model obtained by Levy collapsing a weakly compact (supercompact) cardinal to a>2, first countable ¡»^-collectionwise Hausdorff spaces are weakly ^-collectionwise Hausdorff (weakly collectionwise Hausdorff). In the last section we show that assuming E% , a certain 0-family of integer-valued functions exists and that in the model obtained by Levy collapsing a supercompact cardinal to u>i . these families do not exist.

On certain star versions of the Scheepers property

“…We now recollect few definitions from [20,33]. (1) A subset S of a space X is said to be separated if there exists a collection {U x : x ∈ S} of disjoint open sets with x ∈ U x for every x ∈ S and the collection…”