×

zbMATH — the first resource for mathematics

On the relative cellularity of Lindelöf subspaces of topological groups. (English) Zbl 0803.54016
Absolute and relative Souslin type cardinal invariants are considered in the paper. If \(Y\) is a subspace of \(X\) and \(\tau\) is an infinite cardinal, denote by \(\text{cel}_ \tau (Y,X)\) the minimal cardinal \(m\) such that for every family \(\gamma\) of \(G_ \tau\)-sets in \(X\) there exists \(\mu \subseteq \gamma\) with \(|\mu| \leq m\) and \(Y\cap \bigcup \gamma \subseteq \text{cl} (\bigcup\mu)\). The cardinal \(\text{cel}_ \tau (X,X)\) is abbreviated as \(\text{cel}_ \tau (X)\). The main result of the paper is the following
Theorem: If a subspace \(X\) of a topological group \(G\) satisfies \(\ell(X) \leq\tau\), then \(\text{cel}_ \tau (X,G)\leq \exp \tau\). If, in addition, \(X\) is a retract of \(G\), then \(\text{cel}_ \tau X\leq \exp\tau\).
In particular, every topological group \(H\) with \(\ell(H) \leq\tau\) satisfies \(\text{cel}_ \tau (H)\leq \exp\tau\).

MSC:
54C05 Continuous maps
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
22A05 Structure of general topological groups
54D30 Compactness
PDF BibTeX Cite
Full Text: DOI
References:
[1] Arkhangelskii, A.V., Topological homogeneity. topological groups and their continuous images, Uspekhi mat. nauk, Russian math. surveys, 42, 2, 83-131, (1987), (in English) · Zbl 0642.54017
[2] Guran, I.I., On topological groups close to being Lindelöf, Dokl. akad. nauk SSSR, Soviet math. dokl., 23, 173-175, (1981), (in English) · Zbl 0478.22002
[3] Pasynkov, B.A., On τ-cellularity of τ-Lindelöf topological groups, Vestnik moskov. univ. ser. I mat. mekh., 2, 107, (1992), (in Russian)
[4] Shirokov, L.V., An external characterization of dugundji spaces and κ-metrizable compact Hausdorff spaces, Dokl. akad. nauk SSSR, Soviet math. dokl., 25, 507-510, (1982), (in English) · Zbl 0515.54019
[5] Tkačenko, M.G., Some results on inverse spectra II, Comment. math. univ. carolin., 22, 819-841, (1981) · Zbl 0494.54007
[6] Tkačenko, M.G., On topologies of free groups, Czechoslovak math. J., 34, 109, 541-551, (1984) · Zbl 0584.22001
[7] Uspenskii, V.V., A topological group generated by a Lindelöf σ-space has the souslin property, Dokl. akad. nauk SSSR, Soviet math. dokl., 26, 166-169, (1982), (in English) · Zbl 0527.22001
[8] Uspenskii, V.V., Topological groups and dugunji compacta, Mat. sb., Math. USSR-sb., 67, 555-580, (1990), (in English) · Zbl 0702.22002
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.