Discrete selectivity, shrinking properties, and disjoint local π-bases

Abstract

Using binary function spaces, we give an example of a pseudocompact discretely selective topological group. We show that, under PFA,  every compact space of countable tightness has a countable disjoint local π-base at every point. If X is a compact space of countable  tightness and all non-empty open subsets of X are non-separable, then it is proved in ZFC that X possesses a countable disjoint local π- base at every point. We also establish that a Lindel¨of Σ-space has the discrete shrinking property if and only if the outer π-character of  any compact subset of X is uncountable. As a consequence, any non-metrizable topological group with a countable network has the  discrete shrinking property.