×

Linear subspace of \(\mathbb{R}^ \lambda\) without dense totally disconnected subsets. (English) Zbl 0808.54024

Summary: In [ibid. 125, 231-235 (1985; Zbl 0589.54031)] the author showed that if there is a cardinal \(\kappa\) such that \(2^ \kappa = \kappa^ +\) then there exists a completely regular space without dense 0-dimensional subspaces. This was a solution of a problem of Arkhangel’skij. Recently Arkhangel’skij asked the author whether one can generalize this result by constructing a completely regular space without dense totally disconnected subspaces, and whether such a space can have a structure of a linear space. The purpose of this paper is to show that indeed such a space can be constructed under the additional assumption that there exists a cardinal \(\kappa\) such that \(2^ \kappa = \kappa^ +\) and \(2^{\kappa^ +} = \kappa^{++}\).

MSC:

54G20 Counterexamples in general topology
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)

Citations:

Zbl 0589.54031
PDFBibTeX XMLCite
Full Text: DOI EuDML