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- W4286967758 abstract "It is an interesting, maybe surprising, fact that different dense subspaces of even nice topological spaces can have different densities. So, our aim here is to investigate the set of densities of all dense subspaces of a topological space $X$ that we call the double density spectrum of $X$ and denote by $dd(X)$. We improve a result of Berner and Juhasz by showing that $dd(X)$ is always $omega$-closed (i.e. countably closed) if $X$ is Hausdorff. We manage to give complete characterizations of the double density spectra of Hausdorff and of regular spaces as follows. Let $S$ be a non-empty set of infinite cardinals. Then (1) $S = dd(X)$ holds for a Hausdorff space $X$ iff S is $omega$-closed and $sup S le 2^{2^{min S}},$ (2) S = dd(X) holds for a regular space X iff S is $omega$-closed and $sup S le {2^{min S}}$. We also prove a number of consistency results concerning the double density spectra of compact spaces. For instance: (i) If $kappa = cf(kappa)$ embeds in $mathcal{P}(omega)/fin$ and $S$ is any set of uncountable regular cardinals $< kappa$ with $|S| < min S$, then there is a compactum $C$ such that ${omega, kappa} cup S subset dd(C)$, moreover $lambda notin d(C)$ whenever $|S| + omega < cf(lambda) < kappa$ and $cf(lambda) notin S$. (ii) It is consistent to have a separable compactum $C$ such that $dd(C)$ is not $omega_1$-closed." @default.
- W4286967758 created "2022-07-25" @default.
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- W4286967758 date "2021-09-22" @default.
- W4286967758 modified "2023-10-14" @default.
- W4286967758 title "The double density spectrum of a topological space" @default.
- W4286967758 doi "https://doi.org/10.48550/arxiv.2109.10823" @default.
- W4286967758 hasPublicationYear "2021" @default.
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