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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 \(\text{cel}_ \tau (H)\leq \exp\tau\).

54C05 Continuous maps
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
22A05 Structure of general topological groups
54D30 Compactness
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