Ciesielski, Krzysztof Linear subspace of \(\mathbb{R}^ \lambda\) without dense totally disconnected subsets. (English) Zbl 0808.54024 Fundam. Math. 142, No. 1, 85-88 (1993). 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 \(2^{\kappa^ +} = \kappa^{++}\). Cited in 1 Document MSC: 54G20 Counterexamples in general topology 54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets) Keywords:completely regular space; dense totally disconnected subspaces Citations:Zbl 0589.54031 PDFBibTeX XMLCite \textit{K. Ciesielski}, Fundam. Math. 142, No. 1, 85--88 (1993; Zbl 0808.54024) Full Text: DOI EuDML