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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 paracompactness and property $\left(a\right)$. It follows that it is consistent that closed discrete subsets of a separable space $X$ which are also relatively normal (relatively countably paracompact, relatively $\left(a\right)$) in $X$ are necessarily countable. There are, however, consistent examples of...

Generalizations of earlier negative results on Property $\left(a\right)$ are proved and two questions on an $\left(a\right)$-version of Jones’ Lemma are posed. We discuss these questions in the realm of locally compact spaces. Using dominating families of functions as a tool, we prove that under the assumptions “$2^{\omega }$ is regular” and “$2^{\omega }<2^{{\omega }_1}$” the existence of a $T_1$ separable locally compact $\left(a\right)$-space with an uncountable closed discrete subset implies the existence of inner models with measurable cardinals. We also use cardinal invariants...

We define two cardinal invariants of the continuum which arise naturally from combinatorially and topologically appealing properties of almost disjoint families of sets of the natural numbers. These are the never soft and never countably paracompact numbers. We show that these cardinals must both be equal to ${\omega }_1$ under the effective weak diamond principle $♦(\omega ,\omega ,<)$, answering questions of da Silva S.G., On the presence of countable paracompactness, normality and property $\left(a\right)$ in spaces from almost disjoint families,...