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

Süleyman Önal; Çetin Vural

Open Mathematics (2013)

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

Abstract

top
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

top

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

top
  1. [1] Collins P.J., Reed G.M., Roscoe A.W., Rudin M.E., A lattice of conditions on topological spaces, Proc. Amer. Math. Soc., 1985, 94(3), 487–496 http://dx.doi.org/10.1090/S0002-9939-1985-0787900-X Zbl0562.54043
  2. [2] Erdős P., Hajnal A., Máté A., Rado R., Combinatorial Set Theory: Partition Relations for Cardinals, Stud. Logic Found. Math., 106, North-Holland, Amsterdam-New York-Oxford, 1984 Zbl0573.03019
  3. [3] Gartside P.M., Moody P.J., Well-ordered (F) spaces, Topology Proc., 1992, 17, 111–130 Zbl0797.54038
  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

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.