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Linear subspace of Rl without dense totally disconnected subsets

K. Ciesielski (1993)

Fundamenta Mathematicae

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...

Lonely points revisited

Jonathan L. Verner (2013)

Commentationes Mathematicae Universitatis Carolinae

In our previous paper, we introduced the notion of a lonely point, due to P. Simon. A point p X is lonely if it is a limit point of a countable dense-in-itself set, it is not a limit point of a countable discrete set and all countable sets whose limit point it is form a filter. We use the space 𝒢 ω from a paper of A. Dow, A.V. Gubbi and A. Szymański [Rigid Stone spaces within ZFC, Proc. Amer. Math. Soc. 102 (1988), no. 3, 745–748] to construct lonely points in ω * . This answers the question of P. Simon...

Luzin and anti-Luzin almost disjoint families

Judith Roitman, Lajos Soukup (1998)

Fundamenta Mathematicae

Under M A ω 1 every uncountable almost disjoint family is either anti-Luzin or has an uncountable Luzin subfamily. This fails under CH. Related properties are also investigated.

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