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^{++}.

**Last modified April 24, 1999.**