Linear subspace of Rl without dense totally disconnected subsets

Volume 142 / 1993

K. Ciesielski Fundamenta Mathematicae 142 (1993), 85-88 DOI: 10.4064/fm-142-1-85-88


In [1] the author showed that if there is a cardinal κ such that $2^κ=κ^+$ then there exists a completely regular space without dense 0-dimensional subspaces. This was a solution of a problem of Arkhangel'ski{ĭ}. Recently Arkhangel'skiĭ 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 κ such that $2^κ=κ^+$ and $2^{κ^+}=κ^{++}$.


  • K. Ciesielski

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