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Keywords:
Lindelöf $p$-group; homogeneous space; Lindelöf $\Sigma $-space; dyadic compactum; countable tightness; $\sigma $-compact; $cdc$-group; $p$-space
Lindelöf $p$-group; homogeneous space; Lindelöf $\Sigma $-space; dyadic compactum; countable tightness; $\sigma $-compact; $cdc$-group; $p$-space
Summary:
It is shown that there exists a $\sigma $-compact topological group which cannot be represented as a continuous image of a Lindelöf $p$-group, see Example 2.8. This result is based on an inequality for the cardinality of continuous images of Lindelöf $p$-groups (Theorem 2.1). A closely related result is Corollary 4.4: if a space $Y$ is a continuous image of a Lindelöf $p$-group, then there exists a covering $\gamma $ of $Y$ by dyadic compacta such that $|\gamma |\leq 2^\omega $. We also show that if a homogeneous compact space $Y$ is a continuous image of a $cdc$-group $G$, then $Y$ is a dyadic compactum (Corollary 3.11).
References:
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