connected topological group is generated by any neighborhood of identity
Theorem  Let $G$ be a connected topological group^{} and $e$ its identity element^{}. If $U$ is any open neighborhood of $e$, then $G$ is generated by $U$.
$$
Proof: Let $U$ be an open neighborhood of $e$. For each $n\in \mathbb{N}$ we denote by ${U}^{n}$ the set of elements of the form ${u}_{1}\mathrm{\dots}{u}_{n}$, where each ${u}_{i}\in U$. Let $W:={\bigcup}_{n\in \mathbb{N}}{U}^{n}$.
Since each ${U}^{n}$ is open (see this entry (http://planetmath.org/BasicResultsInTopologicalGroups)  3), we have that $W$ is an open set. We now see that it is also closed.
Let $g\in \overline{W}$, the closure of $W$. Since $g{U}^{1}$ is an open neighborhood of $g$, it must intersect $W$. Thus, let $h\in W\cap g{U}^{1}$.

•
Since $h\in g{U}^{1}$, then $h=g{u}^{1}$ for some element $u\in U$.

•
Since $h\in W$, then $h\in {U}^{n}$ for some $n\in \mathbb{N}$, i.e. $h={u}_{1}\mathrm{\dots}{u}_{n}$ with each ${u}_{i}\in U$.
We then have $g={u}_{1}\mathrm{\dots}{u}_{n}u$, i.e. $g\in {U}^{n+1}\subseteq W$. Hence, $W$ is closed.
Since $G$ is connected and $W$ is open and closed, we must have $W=G$. This means that $G$ is generated by $U$. $\mathrm{\square}$
Title  connected topological group is generated by any neighborhood of identity 

Canonical name  ConnectedTopologicalGroupIsGeneratedByAnyNeighborhoodOfIdentity 
Date of creation  20130322 18:01:45 
Last modified on  20130322 18:01:45 
Owner  asteroid (17536) 
Last modified by  asteroid (17536) 
Numerical id  7 
Author  asteroid (17536) 
Entry type  Theorem 
Classification  msc 22A05 