The double density spectrum of a topological space (Q6165175)

A subspace of a separable space need not be separable (I leave it to the reader to come up with an example). This is arguably the first instance one learns in general topology of a cardinal function that is not monotone. What sounds even more remarkable is that a dense subspace of a separable space need not be separable itself.\par The authors address the question how varied this phenomenon may be by introducing the \emph{double density spectrum} \(\mathrm{dd}(X)\) of a space~\(X\) as the set of cardinals \(\{d(Y):Y\in\mathcal{D}(X)\}\), where \(\mathcal{D}(X)\) denotes the family of dense subsets of~\(X\). Clearly \(d(X)=\min\mathrm{dd}(X)\) and the well-known inequalities \(|X|\le2^{2^{d(X)}}\) and \(|X|\le2^{d(X)}\) for Hausdorff and regular spaces respectively give definite lower and upper bounds for~\(\mathrm{dd}(X)\). The first non-trivial result is that \(\mathrm{dd}(X)\) is always \(\omega\)-closed: the supremum of a countable subset of~\(\mathrm{dd}(X)\) is again in~\(\mathrm{dd}(X)\).\par For Hausdorff and regular spaces the above inequalities and \(\omega\)-closedness characterize their double density spectra: if \(S\)~is an \(\omega\)-closed set of cardinals and \(\sup S\le2^{2^{\min S}}\) (or \(\sup S\le2^{\min S}\)) then there is a Hausdorff (or regular) space~\(X\) with \(S=\mathrm{dd}(X)\).\par The third section of the paper begins the investigation of the double density spectrum of compact Hausdorff spaces. The results are mostly about separable compacta and involve the smallest cardinal~\(\mathfrak{n}\) that cannot be embedded (order-isomorphically) into \(\mathcal{P}(\omega)/\mathrm{fin}\). A sample result: if \(\mathfrak{n}>\aleph_\omega\) then one can find for every subset~\(a\) of \(\omega\) a separable compact space~\(C_a\) such that \(\mathrm{dd}(C_a)\) is equal to \(\bigl\{\aleph_n:n\in \{0\}\cup a\bigr\}\) if \(a\)~is finite and to \(\bigl\{\aleph_n:n\in \{0\}\cup a\cup\{\omega\}\bigr\}\) if \(a\)~is infinite.