Further applications of bornological covering properties in function spaces

Abstract

This article is a follow up of the work done in the function space C(X) with respect to the topology τBs of strong uniform convergence on B, carried out in (Chandra et al., 2020 [12]) using the idea of strong uniform convergence (Beer and Levi, 2009 [6]) on a bornology. We present characterizations of various local properties of (C(X),τBs) such as countable fan tightness for finite sets, countable strong fan tightness for finite sets, Fréchet-Urysohn for finite sets etc. in terms of selection principles related to bornological covers of X (represented by a diagram describing various implications among them). Lastly, taking cue from [39,40], we also investigate topological properties of sequences of dense and sequentially dense subsets of (C(X),τBs) and present a diagram describing implications among those selective properties of (C(X),τBs).