Abstract:It is shown that there exists a $\sigma $-compact topological group which cannot be represented as a continuous image of a Lindel\"of $p$-group, see Example~2.8. This result is based on an inequality for the cardinality of continuous images of Lindel\"of $p$-groups (Theorem 2.1). A closely related result is Corollary~4.4: if a~space $Y$ is a continuous image of a Lindel\"of $p$-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).
Keywords: Lindel\"of $p$-group; homogeneous space; Lindel\"of $\Sigma $-space; dyadic compactum; countable tightness; $\sigma $-compact; $cdc$-group; $p$-space
DOI: DOI 10.14712/1213-7243.2019.027
AMS Subject Classification: 54A25 54B05