# Showing that the map $G \rightarrow \ lim_{\leftarrow} G/U$ is open

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 $G$.
• I thought the definition for the topology was that $lim G/U \rightarrow G/U$ is continuous for each $U$? Is this equivalent to what you wrote here? Thanks. – Johnny T. Nov 22 '17 at 21:34