# Linear subspace of Rl without dense totally disconnected subsets

Fundamenta Mathematicae (1993)

- Volume: 142, Issue: 1, page 85-88
- ISSN: 0016-2736

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topCiesielski, K.. "Linear subspace of Rl without dense totally disconnected subsets." Fundamenta Mathematicae 142.1 (1993): 85-88. <http://eudml.org/doc/211973>.

@article{Ciesielski1993,

abstract = {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’skiĭ. 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 $2^\{κ^+\}=κ^\{++\}$.},

author = {Ciesielski, K.},

journal = {Fundamenta Mathematicae},

keywords = {completely regular space; dense totally disconnected subspaces},

language = {eng},

number = {1},

pages = {85-88},

title = {Linear subspace of Rl without dense totally disconnected subsets},

url = {http://eudml.org/doc/211973},

volume = {142},

year = {1993},

}

TY - JOUR

AU - Ciesielski, K.

TI - Linear subspace of Rl without dense totally disconnected subsets

JO - Fundamenta Mathematicae

PY - 1993

VL - 142

IS - 1

SP - 85

EP - 88

AB - 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’skiĭ. 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 $2^{κ^+}=κ^{++}$.

LA - eng

KW - completely regular space; dense totally disconnected subspaces

UR - http://eudml.org/doc/211973

ER -

## References

top- [1] K. Ciesielski, L-space without any uncountable 0-dimensional subspace, Fund. Math. 125 (1985), 231-235. Zbl0589.54031
- [2] R. Engelking, General Topology, Polish Scientific Publishers, Warszawa 1977.
- [3] K. Kunen, Set Theory, North-Holland, 1983.

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