Suppose $G$ is compact (Hausdorff topological group) and the unit elements has a basis of neighbourhoods consisting of open and closed normal subgroups. Let $U$ run through a system of neighbourhoods of the unit element which consists of open normal subgroups.
How does one show that the canonical projection $G \rightarrow \ lim_{\leftarrow} G/U$ (inverse limit) is open? In the book I am reading it is stated without any explanation and I would greatly appreciate some explanation. Thank you.
By definition, $lim G/U$ is endowed with the quotient topology whose class of open subspaces is generated by the family of $p(V)$ where $p:G\rightarrow lim G/U$ is the quotient map and $V$ is open in $G$.