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

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