# 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 $u_{1}\dots u_{n}$, 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 $gU^{-1}$ is an open neighborhood of $g$, it must intersect $W$. Thus, let $h\in W\cap gU^{-1}$.

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

• Since $h\in W$, then $h\in U^{n}$ for some $n\in\mathbb{N}$, i.e. $h=u_{1}\dots u_{n}$ with each $u_{i}\in U$.

We then have $g=u_{1}\dots u_{n}u$, 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 $U$. $\square$

Title connected topological group is generated by any neighborhood of identity ConnectedTopologicalGroupIsGeneratedByAnyNeighborhoodOfIdentity 2013-03-22 18:01:45 2013-03-22 18:01:45 asteroid (17536) asteroid (17536) 7 asteroid (17536) Theorem msc 22A05