Closed discrete subsets of separable spaces and relative versions of normality, countable paracompactness and property (a)

In this paper we show that a separable space cannot include closed discrete subsets which have the cardinality of the continuum and satisfy relative versions of any of the following topological properties: normality, countable and property (a). It follows that it is consistent that closed discrete subsets of a eparable space A' which are aX relatively normal (relatively countably paracompact, relatively (a)) in A' are necessaXly countable. There are, however, consistent examples of separable spaces with uncountable closed discrete subsets under the described relative topological requirements, and therefore the existenceof such uncountable sets is undecidable within ZFC. We also investigate what are the outcomes of considering the set-theoretical hypothesis "2(W) < 2(W1)" within our(omega)discussion and conclude by giving some notes and posing some questions.