Connected subspace, connected subgroup.

Question

I'm studying for my Introductory topology class and I came up with a question that I haven't been able to answer.

Problem: Let's suppose we have a connected subspace $C$ of a topological group $G$ that contains the identity element. Now let's call $Z$ the subgroup generated by $C$. Is $Z$ connected? I guess it is but I haven't been able to show this fact.

Any resources, proofs, etc. are welcome.

Thanks so much for your help guys, I really appreciate it.

Answer 1

Given a set $S$, call $S_n=\{u\,:\, \exists (g_1,\cdots, g_n)\in S^n, u=g_1\cdots g_n\}$. Call $C$ your initial set and call $S=C\cup C^{-1}$, where $C^{-1}=\{u^{-1}\,:\, u\in C\}$. Notice that $S$ is connected because $C$ and $C^{-1}$ are, and $C\cap C^{-1}\supseteq \{e\}\supsetneq\emptyset$. Moreover, again because $e\in S$, we have that $S_{n+1}\supseteq S_n$ for all $n$. Also, $S_n$ is connected for all $n$. This can be proved inductively, because $S_{n+1}$ is the image by multiplication of the connected set $S\times S_n$. Finally, you know that $Z=\bigcup_{n\ge 1} S_n$. Therefore this is the union of an increasing sequence of connected subsets, which is connected.