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

Andrey V. Koldunov; Aleksandr I. Veksler

Commentationes Mathematicae Universitatis Carolinae (2001)

  • Volume: 42, Issue: 2, page 363-378
  • ISSN: 0010-2628

Abstract

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In [5] the following question was put: are there any maximal n.d. sets in ω * ? Already in [9] the negative answer (under 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 CH there is no such a set in ω * . 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.

How to cite

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Koldunov, Andrey V., and Veksler, Aleksandr I.. "Maximal nowhere dense $P$-sets in basically disconnected spaces and $F$-spaces." Commentationes Mathematicae Universitatis Carolinae 42.2 (2001): 363-378. <http://eudml.org/doc/248818>.

@article{Koldunov2001,
abstract = {In [5] the following question was put: are there any maximal n.d. sets in $\omega ^*$? Already in [9] the negative answer (under 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 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.},
author = {Koldunov, Andrey V., Veksler, Aleksandr I.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {maximal n.d. set; $P$-set; maximal n.d. $P$-set; compact space; basically disconnected space; $F$-space; nowhere dense set; -set; compact space; basically disconnected; -space},
language = {eng},
number = {2},
pages = {363-378},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Maximal nowhere dense $P$-sets in basically disconnected spaces and $F$-spaces},
url = {http://eudml.org/doc/248818},
volume = {42},
year = {2001},
}

TY - JOUR
AU - Koldunov, Andrey V.
AU - Veksler, Aleksandr I.
TI - Maximal nowhere dense $P$-sets in basically disconnected spaces and $F$-spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2001
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 42
IS - 2
SP - 363
EP - 378
AB - In [5] the following question was put: are there any maximal n.d. sets in $\omega ^*$? Already in [9] the negative answer (under 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 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.
LA - eng
KW - maximal n.d. set; $P$-set; maximal n.d. $P$-set; compact space; basically disconnected space; $F$-space; nowhere dense set; -set; compact space; basically disconnected; -space
UR - http://eudml.org/doc/248818
ER -

References

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  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. (1984) MR0770230
  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. (1991) Zbl0737.54010MR1109658
  13. Vermeer J., The smallest basically disconnected preimage of a space, Topology Appl. 17:3 (1984), 217-232. (1984) Zbl0593.54036MR0752272
  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. (1980) MR0581394
  15. Zakharov V.K., Koldunov A.V., Characterization of the σ -covering of compactum (in Russian), Sibirsk. Mat. Zh. 23:6 (1982), 91-99; English transl.: Siberian Math. J. 23:6 (1982), 834-851. (1982) MR0682910

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