Locally invariant topological groups and semidirect products

Abstract:

We consider topological groups which have arbitrarily small invariant neighborhoods of the identity and those topological groups which admit continuous monomorphisms into such groups. We establish conditions under which the two corresponding classes of groups coincide. We apply these results to semidirect products. Since we do not assume local compactness in general, we use the symbol "[Sn]" rather than "[SIN]" for the class of groups with small invariant neighborhoods and the symbol "[In]" for those embeddable in Sn groups. We denote by "[N]" those groups $G$ with the property: If ${\{ {x_\alpha }\} _{\alpha \in D}}$ is a net in $G$ which converges to the identity and ${\{ {y_\alpha }\} _{\alpha \in D}}$ is any net such that $\{ {y_\alpha }{x_\alpha }y_\alpha ^{ - 1}\}$ converges, then this net also converges to the identity. We also define a class of topological groups we term $S(U)$ groups. The following are corollaries of our general results: (1) If $G$ is locally compact, $G/{G_0}$ is compact and ${G_0}$ is an $N$ groups, then $G$ is an Sn group. (2) If $H$ is a locally connected compact group, $G$ is an Sn group, and if the semidirect product $H\circledS G$ is an $S(U)$ group, then $H\circledS G$ is an Sn group. (3) If $H\circledS K$ is an Sn group for every compact group $K$, then every open subgroup of $H$ is of finite index.