# Article

Full entry | PDF   (0.2 MB)
Keywords:
maximal n.d. set; $P$-set; maximal n.d. $P$-set; compact space; basically disconnected space; $F$-space
Summary:
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.
References:
[1] 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). MR 0472520
[2] Henriksen M., Vermeer J., Woods R.G.: Quasi-$F$-covers of Tychonoff space. Trans. Amer. Math. Soc. 303:2 (1987), 779-803. MR 0902798
[3] Koldunov A.V.: $\sigma$-completion and $0$-completion of $C(B)$ (in Russian). Functional Analysis (Uljanovsk), no. 6 (1976), 76-83. MR 0626059
[4] 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. MR 0579882
[5] van Mill J., Reed E.M. (eds.): Open Problems in Topology. North-Holland Co., Amsterdam, 1990. MR 1078636 | Zbl 0877.54001
[6] Park Y.L.: The Quasi-$F$-cover as a filter space. Houston J. Math. 9:1 (1983), 101-109. MR 0699052
[7] Simon P.: A note on nowhere dense sets in $ømega ^*$. Comment. Math. Univ. Carolinae 31:1 (1990), 145-147. MR 1056181 | Zbl 0696.54019
[8] Simon P.: A note on almost disjoint refinement. Acta Univ. Carolinae, Math. et Phys. 37:2 (1996), 89-99. MR 1600453 | Zbl 0883.04003
[9] Veksler A.I.: Maximal nowhere dense sets in topological spaces (in Russian). Izvestiya VUZ, Mathem. no. 5 (1975), 9-16. MR 0410705
[10] Veksler A.I.: $P'$-sets in regular spaces. Colloquia Math. Soc. J. Bolyai 23 Topology, Budapest, pp.1211-1225 (1978). MR 0588868
[11] 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. MR 0770230
[12] 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. MR 1109658 | Zbl 0737.54010
[13] Vermeer J.: The smallest basically disconnected preimage of a space. Topology Appl. 17:3 (1984), 217-232. MR 0752272 | Zbl 0593.54036
[14] 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. MR 0581394
[15] 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. MR 0682910

Partner of