R-spaces and closedness/completeness of certain function spaces in the topology of uniform convergence

Abstract

It is shown that the notion of an R cl -space (Demonstratio Math. 46(1) (2013), 229-244) fits well as a separation axiom between zero dimensionality and R0-spaces. Basic properties of R cl -spaces are studied and their place in the hierarchy of separation axioms that already exist in the literature is elaborated. The category of R cl -spaces and continuous maps constitutes a full isomorphism closed, monoreflective (epireflective) subcategory of TOP. The function space R cl (X, Y) of all R cl -supercontinuous functions from a space X into a uniform space Y is shown to be closed in the topology of uniform convergence. This strengthens and extends certain results in the literature (Demonstratio Math. 45(4) (2012), 947-952).2010 MSC: 54C08; 54C10; 54C35; 54D05; 54D10.