Linear subspace of Rlambda without dense totally disconnected subset

by

Krzysztof Ciesielski

Fund. Math. 142 (1993), 85-88

In [K. Ciesielski, L-spaces without any uncountable 0-dimensional subspace, Fund. Math. 125 (1985), 231-235] 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 subspace. This was a solution of a problem of Arhangiel'skii. Recently Arhangiel'skii asked the author (private communication) whether we can generalize this result by constructing a completely regular space without dense totally disconnected subspace, 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++.