# On weaker forms of the chain (F) condition and metacompactness-like covering properties in the product spaces

Open Mathematics (2013)

• Volume: 11, Issue: 9, page 1635-1642
• ISSN: 2391-5455

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## Abstract

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We introduce the concept of a family of sets generating another family. Then we prove that if X is a topological space and X has W = {W(x): x ∈ X} which is finitely generated by a countable family satisfying (F) which consists of families each Noetherian of ω-rank, then X is metaLindelöf as well as a countable product of them. We also prove that if W satisfies ω-rank (F) and, for every x ∈ X, W(x) is of the form W 0(x) ∪ W 1(x), where W 0(x) is Noetherian and W 1(x) consists of neighbourhoods of x, then X is metacompact.

## How to cite

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Süleyman Önal, and Çetin Vural. "On weaker forms of the chain (F) condition and metacompactness-like covering properties in the product spaces." Open Mathematics 11.9 (2013): 1635-1642. <http://eudml.org/doc/269192>.

@article{SüleymanÖnal2013,
abstract = {We introduce the concept of a family of sets generating another family. Then we prove that if X is a topological space and X has W = \{W(x): x ∈ X\} which is finitely generated by a countable family satisfying (F) which consists of families each Noetherian of ω-rank, then X is metaLindelöf as well as a countable product of them. We also prove that if W satisfies ω-rank (F) and, for every x ∈ X, W(x) is of the form W 0(x) ∪ W 1(x), where W 0(x) is Noetherian and W 1(x) consists of neighbourhoods of x, then X is metacompact.},
author = {Süleyman Önal, Çetin Vural},
journal = {Open Mathematics},
keywords = {Metacompact; MetaLindelöf; Product spaces; Noetherian; Rank; metacompact; metaLindelöf; product spaces; rank},
language = {eng},
number = {9},
pages = {1635-1642},
title = {On weaker forms of the chain (F) condition and metacompactness-like covering properties in the product spaces},
url = {http://eudml.org/doc/269192},
volume = {11},
year = {2013},
}

TY - JOUR
AU - Süleyman Önal
AU - Çetin Vural
TI - On weaker forms of the chain (F) condition and metacompactness-like covering properties in the product spaces
JO - Open Mathematics
PY - 2013
VL - 11
IS - 9
SP - 1635
EP - 1642
AB - We introduce the concept of a family of sets generating another family. Then we prove that if X is a topological space and X has W = {W(x): x ∈ X} which is finitely generated by a countable family satisfying (F) which consists of families each Noetherian of ω-rank, then X is metaLindelöf as well as a countable product of them. We also prove that if W satisfies ω-rank (F) and, for every x ∈ X, W(x) is of the form W 0(x) ∪ W 1(x), where W 0(x) is Noetherian and W 1(x) consists of neighbourhoods of x, then X is metacompact.
LA - eng
KW - Metacompact; MetaLindelöf; Product spaces; Noetherian; Rank; metacompact; metaLindelöf; product spaces; rank
UR - http://eudml.org/doc/269192
ER -

## References

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4. [4] Gruenhage G., Nyikos P., Spaces with bases of countable rank, General Topology and Appl., 1978, 8(3), 233–257 http://dx.doi.org/10.1016/0016-660X(78)90004-1 Zbl0412.54034
5. [5] Hajnal A., Hamburger P., Set Theory, London Math. Soc. Stud. Texts, 48, Cambridge University Press, Cambridge, 1999 http://dx.doi.org/10.1017/CBO9780511623561
6. [6] Moody P.J., Reed G.M., Roscoe A.W., Collins P.J., A lattice of conditions on topological spaces. II, Fund. Math., 1991, 138(2), 69–81 Zbl0745.54008
7. [7] Nagata J., On dimension and metrization, In: General Topology and its Relations to Modern Analysis and Algebra, Prague, 1961, Academic Press, New York, 1962, 282–285
8. [8] Nyikos P.J., Some surprising base properties in topology. II, In: Set-Theoretic Topology, Athens, 1975–1976, Academic Press, New York-London, 1977, 277–305
9. [9] Vural Ç., Some weaker forms of the chain (F) condition for metacompactness, J. Aust. Math. Soc., 2008, 84(2), 283–288 http://dx.doi.org/10.1017/S1446788708000037 Zbl1151.54020

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