$(\mathbb{R}^n,d)$ metric space, $U \subseteq \mathbb{R}^n$ connected and open subset. Show that $U$ is also path-connected

Asked Jan 21 '20 at 12:50

Active Jan 21 '20 at 12:50

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I'm working on the following exercise:

Let $X=\mathbb{R}^n$ with the metric topology and let $U \subseteq \mathbb{R}^n$ be a connected and open subset. Show that $U$ is also path-connected.

It's kind of obvious, but I haven't been able to prove it.

asked Jan 21 '20 at 12:50

Tom

1

You may look here https://math.stackexchange.com/questions/332108/showing-that-every-connected-open-set-in-a-locally-path-connected-space-is-path?rq=1 – Severin Schraven Jan 21 '20 at 12:53
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Hint: show that the path-connected components of $U$ are open. – Aphelli Jan 21 '20 at 13:01

$(\mathbb{R}^n,d)$ metric space, $U \subseteq \mathbb{R}^n$ connected and open subset. Show that $U$ is also path-connected

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