Maximal nowhere dense $P$-sets in basically disconnected spaces and $F$-spaces

Koldunov, A. V.; Veksler, A. I.

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Maximal nowhere dense $P$-sets in basically disconnected spaces and $F$-spaces

Koldunov, A. V. ; Veksler, A. I.

Commentationes Mathematicae Universitatis Carolinae, Tome 42 (2001) no. 2, pp. 363-378.

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Résumé

In [5] the following question was put: are there any maximal n.d. sets in $\omega^*$? Already in [9] the negative answer (under {\bf MA}) to this question was obtained. Moreover, in [9] it was shown that no $P$-set can be maximal n.d. In the present paper the notion of a maximal n.d. $P$-set is introduced and it is proved that under {\bf CH} there is no such a set in $\omega^*$. The main results are Theorem 1.10 and especially Theorem 2.7(ii) (with Example in Section 3) in which the problem of the existence of maximal n.d. $P$-sets in basically disconnected compact spaces with rich families of n.d. $P$-sets is actually solved.

MR Zbl

Classification : 54B05, 54D30, 54D40, 54G05
Keywords: maximal n.d. set; $P$-set; maximal n.d. $P$-set; compact space; basically disconnected space; $F$-space

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     title = {Maximal nowhere dense $P$-sets in basically disconnected spaces and $F$-spaces},
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Koldunov, A. V.; Veksler, A. I. Maximal nowhere dense $P$-sets in basically disconnected spaces and $F$-spaces. Commentationes Mathematicae Universitatis Carolinae, Tome 42 (2001) no. 2, pp. 363-378. http://geodesic.mathdoc.fr/item/CMUC_2001__42_2_a13/