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 $U$.

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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	2013-03-22 18:01:45
Last modified on	2013-03-22 18:01:45
Owner	asteroid (17536)
Last modified by	asteroid (17536)
Numerical id	7
Author	asteroid (17536)
Entry type	Theorem
Classification	msc 22A05