×

\(D\)-spaces and covering properties. (English) Zbl 1063.54013

A neighbourhood assignment on a topological space is a mapping \(\phi\) of \(X\) into the topology \(\tau\) of \(X\) such that \(x\in \phi(x)\) for each \(x \in X\). A space \(X\) is a D-space if, for every neighbourhood assignment \(\phi\) on \(X\), there exists a locally finite (in \(X\)) subset \(A\) of \(X\) such that the family \(\phi(A)\) covers \(X\).
This paper is a continuation of the author’s earlier paper (with R. Z. Buzyakova) [Commentat. Math. Univ. Carol. 43, 653–663 (2002)]. We quote the author’s abstract:
We study the D-space property and its generalizations, the notions of an aD-space and a weak aD-space in connection with covering properties. A brief survey on D-spaces is presented in Section 1.
Among new results, it is proved that if a linearly ordered space is an aD-space, then it is paracompact. The statement further extends the list of equivalences in [E. K. van Douwen and D. J. Lutzer, Proc. Am. Math. Soc. 125, 1237–1245 (1997; Zbl 0885.54023)]. We also establish some sufficient conditions for the free topological group of a Tychonoff space to be a D-space. In particular, the free topological group of a semi-stratifiable space is shown to be a D-space, while it need not be semi-stratifiable. A similar result is established for the free topological group of a space with a point-countable base. Some new interesting open problems on D-spaces and on spaces close to them are formulated. In particular, we discuss several such questions in connection with the sum theorems.

MSC:

54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54J05 Nonstandard topology

Citations:

Zbl 0885.54023
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Arens, R.; Dugundji, J., Remark on the concept of compactness, Portugal. Math, 9, 141-143 (1950) · Zbl 0039.18602
[2] Arhangel’skii, A. V., On relations between invariants of topological groups and their subspaces, Russian Math. Surveys, 35, 3, 3-22 (1980)
[3] Arhangel’skii, A. V.; Buzyakova, R. Z., Addition theorems and \(D\)-spaces, Comment. Math. Univ. Carolin, 43, 4, 653-663 (2002) · Zbl 1090.54017
[4] Aull, C. E., A generalization of a theorem of Aquaro, Bull. Austral. Math. Soc, 9, 105-108 (1973) · Zbl 0255.54015
[5] Baturov, D. P., Subspaces of function spaces, Moscow Univ. Math. Bull, 42, 4, 75-78 (1987) · Zbl 0665.54004
[6] Bennett, H. R.; Lutzer, D. J., A note on weak \(θ\)-refinability, Gen. Topology Appl, 2, 49-54 (1972) · Zbl 0229.54022
[7] Boone, J. R., On irreducible spaces, Bull. Austral. Math. Soc, 12, 143-148 (1975) · Zbl 0285.54013
[8] Boone, J. R., On irreducible spaces. 2, Pacific J. Math, 62, 2, 351-357 (1976) · Zbl 0327.54012
[9] Borges, C. R.; Wehrly, A. C., A study of \(D\)-spaces, Topology Proc, 16, 7-15 (1991) · Zbl 0787.54023
[10] Burke, D. K., A note on R.H. Bing’s example \(G\), (Proc. V.P.I. Topology Conference (1974), Springer: Springer Berlin)
[11] Burke, D. K., Covering properties, (Kunen, K.; Vaughan, J., Handbook of Set-Theoretic Topology (1984), North-Holland: North-Holland Amsterdam), 347-422, Chapter 9
[12] Buzyakova, R. Z., On \(D\)-property of strong \(Σ\)-spaces, Comment. Math. Univ. Carolin, 43, 3, 493-495 (2002) · Zbl 1090.54018
[13] R.Z. Buzyakova, Hereditary \(D\); R.Z. Buzyakova, Hereditary \(D\)
[14] Chaber, J., Metacompactness and the class MOBI, Fund. Math, 91, 211-217 (1976) · Zbl 0343.54010
[15] Christian, U., Concerning certain minimal cover refinable spaces, Fund. Math, 76, 213-222 (1972) · Zbl 0251.54011
[16] van Douwen, E.; Lutzer, D. J., A note on paracompactness in generalized ordered spaces, Proc. Amer. Math. Soc, 125, 1237-1245 (1997) · Zbl 0885.54023
[17] van Douwen, E.; Pfeffer, W. F., Some properties of the Sorgenfrey line and related spaces, Pacific J. Math, 81, 2, 371-377 (1979) · Zbl 0409.54011
[18] van Douwen, E. K.; Wicke, H. H., A real, weird topology on reals, Houston J. Math, 13, 1, 141-152 (1977) · Zbl 0345.54036
[19] Dow, A.; Junnila, H.; Pelant, J., Weak covering properties of weak topologies, Proc. London Math. Soc, 75, 3, 349-368 (1997) · Zbl 0886.54014
[20] A. Dow, H. Junnila, J. Pelant, More on weak covering properties of weak topologies, Preprint; A. Dow, H. Junnila, J. Pelant, More on weak covering properties of weak topologies, Preprint · Zbl 0886.54014
[21] Grothendieck, A., Criteres de compacité dans les espaces fonctionnels génereaux, Amer. J. Math, 74, 168-186 (1952) · Zbl 0046.11702
[22] Gruenhage, G., Generalized metric spaces, (Kunen, K.; Vaughan, J., Handbook of Set-Theoretic Topology (1984), North-Holland: North-Holland Amsterdam), 423-501, Chapter 10
[23] Mashburn, J., A note on irreducibility and weak covering properties, Topology Proc, 9, 2, 339-352 (1984) · Zbl 0577.54017
[24] Nagami, K., \(Σ\)-spaces, Fund. Math, 61, 169-192 (1969) · Zbl 0181.50701
[25] Ostaszewski, A. J., Compact \(σ\)-metric spaces are sequential, Proc. Amer. Math. Soc, 68, 339-343 (1978) · Zbl 0392.54014
[26] Rudin, M. E., Dowker spaces, (Kunen, K.; Vaughan, J., Handbook of Set-Theoretic Topology (1984), North-Holland: North-Holland Amsterdam), 761-780, Chapter 17 · Zbl 0566.54009
[27] Tkachenko, M. G., On compactness of countably compact spaces having additional structure, Trans. Mosc. Math. Soc, 2, 149-167 (1984) · Zbl 0557.54016
[28] Worrell, J. M.; Wicke, H. H., A covering property which implies isocompactness. 1, Proc. Amer. Math. Soc, 79, 2, 331-334 (1980) · Zbl 0433.54010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.