Maximal nowhere dense $P$-sets in basically disconnected spaces and $F$-spaces
Koldunov, Andrey V. ; Veksler, Aleksandr I.
Commentationes Mathematicae Universitatis Carolinae, Tome 42 (2001), p. 363-378 / Harvested from Czech Digital Mathematics Library
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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.

Publié le : 2001-01-01
Classification:  54B05,  54D30,  54D40,  54G05 
@article{119251,
     author = {Andrey V. Koldunov and Aleksandr I. Veksler},
     title = {Maximal nowhere dense $P$-sets in basically disconnected spaces and $F$-spaces},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     volume = {42},
     year = {2001},
     pages = {363-378},
     zbl = {1053.54041},
     mrnumber = {1832155},
     language = {en},
     url = {http://dml.mathdoc.fr/item/119251}
}

Koldunov, Andrey V.; Veksler, Aleksandr I. Maximal nowhere dense $P$-sets in basically disconnected spaces and $F$-spaces. Commentationes Mathematicae Universitatis Carolinae, Tome 42 (2001) pp. 363-378. http://gdmltest.u-ga.fr/item/119251/

Hechler S.H. Generalizations of almost disjointness, $c$-sets, and the Baire number of $\beta N\backslash N$, General Topology Appl. 8:1 93-110 (1978). (1978) | MR 0472520

Henriksen M.; Vermeer J.; Woods R.G. Quasi-$F$-covers of Tychonoff space, Trans. Amer. Math. Soc. 303:2 (1987), 779-803. (1987) | MR 0902798

Koldunov A.V. $\sigma$-completion and $0$-completion of $C(B)$ (in Russian), Functional Analysis (Uljanovsk), no. 6 (1976), 76-83. (1976) | MR 0626059

Koldunov A.V. On maximal nowhere dense sets (in Russian), Sibirsk. Mat. Zh. 21:4 (1980), 103-111; English transl.: Siberian Math. J. 21 (1981), 558-563. (1980) | MR 0579882

Van Mill J.; Reed E.M. (Eds.) Open Problems in Topology, North-Holland Co., Amsterdam, 1990. | MR 1078636 | Zbl 0877.54001

Park Y.L. The Quasi-$F$-cover as a filter space, Houston J. Math. 9:1 (1983), 101-109. (1983) | MR 0699052

Simon P. A note on nowhere dense sets in $ømega ^*$, Comment. Math. Univ. Carolinae 31:1 (1990), 145-147. (1990) | MR 1056181 | Zbl 0696.54019

Simon P. A note on almost disjoint refinement, Acta Univ. Carolinae, Math. et Phys. 37:2 (1996), 89-99. (1996) | MR 1600453 | Zbl 0883.04003

Veksler A.I. Maximal nowhere dense sets in topological spaces (in Russian), Izvestiya VUZ, Mathem. no. 5 (1975), 9-16. (1975) | MR 0410705

Veksler A.I. $P'$-sets in regular spaces, Colloquia Math. Soc. J. Bolyai 23 Topology, Budapest, pp.1211-1225 (1978). (1978) | MR 0588868

Veksler A.I. Nettings in topological spaces (in Russian), Izvestiya VUZ, Mathem. no. 3 (1984), 105-114; English transl.: Sov. Math. 28:3, 32-42. (1984) | MR 0770230

Veksler A.I. Maximal nowhere dense sets and their applications to problems of existence of remote points and of weak $P$-points, Math. Nachr. 150 (1991), 263-276. (1991) | MR 1109658 | Zbl 0737.54010

Vermeer J. The smallest basically disconnected preimage of a space, Topology Appl. 17:3 (1984), 217-232. (1984) | MR 0752272 | Zbl 0593.54036

Zakharov V.K.; Koldunov A.V. The sequential absolute and its characterization (in Russian), DAN SSSR 253:2 (1980), 280-284; English transl.: Soviet. Math. Dokl. 22:1 (1980), 70-74. (1980) | MR 0581394

Zakharov V.K.; Koldunov A.V. Characterization of the $\sigma$-covering of compactum (in Russian), Sibirsk. Mat. Zh. 23:6 (1982), 91-99; English transl.: Siberian Math. J. 23:6 (1982), 834-851. (1982) | MR 0682910