Baire category properties of topological groups

Анотація

We present several known and new results on the Baire category properties in topological groups. In particular, we prove that a Baire topological group $X$ is metrizable if and only if $X$ is point-cosmic if and only if $X$ is a $\sigma$-space. A topological group $X$ is Choquet if and only if its Raikov completion $\overline{X}$ is Choquet and $X$ is $G_{\delta }$-dense in $\overline{X}$. A topological group $X$ is complete-metrizable if and only if $X$ is a point-cosmic Choquet space if and only if $X$ is a Choquet $\sigma$-space.  Finally, we pose several open problem, in particular, whether each Choquet topological group is strong Choquet.