How Does a Connected Subspace Generate Another in Topological Groups?

  • MHB
  • Thread starter Euge
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    2015
In summary, POTW #190 is a weekly challenge posted by scientific communities or organizations that presents a problem related to a specific scientific topic for individuals to solve. Its purpose is to encourage critical thinking, problem-solving skills, and collaboration among scientists and individuals interested in the topic. The solution is determined by the posting organization and there may be rewards for solving it, but the main reward is the satisfaction of solving a challenging problem. Anyone with an interest in the topic can participate in POTW #190, regardless of their background or expertise.
  • #1
Euge
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Here is this week's POTW:

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Show that a connected subspace of a topological group $\pi$ which contains the identity must algebraically generate another connected subspace of $\pi$.
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Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!
 
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  • #2
No one answered this week's problem. You can read my solution below.
Let $C$ be a connected subspace of $\pi$ which contains the identity. In particular, the subgroup $\langle C\rangle$ generated by $C$ contains the identity. Let $x$ be a non-identity element of $\langle C\rangle$. There exists an index $k$, elements $c_1,\ldots, c_k\in C$, and positive integers $n_1,\ldots, n_k$ such that $x = c_1^{\epsilon_1 n_1}\cdots c_k^{\epsilon_k n_k}$, where the $\epsilon_i$ are $\pm 1$. Since multiplication and inversion are continuous and $C$ is connected, the sets $C^{\epsilon_1},\ldots, C^{\epsilon_k}$ are connected. Hence $C^{\epsilon_1}\cdots C^{\epsilon_k}$, the continuous image of the connected set $C^{\epsilon_1}\times \cdots \times C^{\epsilon_k}$ (in $\pi^k$) under multiplication, is connected. Since both $x$ and the identity belong to $C^{\epsilon_1}\cdots C^{\epsilon_k}$, it follows that $x$ belongs to the connected component of $\langle C\rangle$ at the identity. Since $x$ was arbitrary, $\langle C\rangle$ is connected.
 

FAQ: How Does a Connected Subspace Generate Another in Topological Groups?

What is POTW #190?

POTW #190 stands for "Problem of the Week #190" and is a weekly challenge posted by various scientific communities or organizations. It presents a problem or puzzle related to a specific scientific topic for individuals to solve.

What is the purpose of POTW #190?

The purpose of POTW #190 is to encourage critical thinking, problem-solving skills, and collaboration among scientists and individuals interested in the specific topic. It also serves as a fun and engaging way to learn and apply scientific concepts.

How is the solution to POTW #190 determined?

The solution to POTW #190 is determined by the scientific community or organization that posted the challenge. They typically provide a set of criteria or guidelines for individuals to follow in order to come up with the correct solution.

Are there any rewards for solving POTW #190?

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