Show that if $A$ is countable subset of $\mathbb{R}^2$, then $\mathbb{R}^2 -A$ is path connected.

Asked Jul 31 '21 at 20:09

Active Jul 31 '21 at 20:09

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I want to prove the following question

Show that if $A$ is countable subset of $\mathbb{R}^2$, then $\mathbb{R}^2 -A$ is path connected.

My attempt:

Let $P, Q \in \mathbb{R}^2 -A$ be any two points. If the unique line passes through $P$ and $Q$ does not intersect $A$, then $\overline{PQ}$ is a path. If not so, there are uncountably many straight lines through $P$, and only countably many of those lines intersect $A$, so there are uncountably many straight lines through $P$ that do not intersect $A$. Similarly, there are uncountably many straight lines through $Q$ that do not intersect $A$. So, there exist a line $l_1$ through $P$ and a line $l_2$ through $Q$ which are not parallel, hence they intersect. Let $\{R\} = l_1 \cap l_2$. Therefore, $\overline{PR} \cup \overline{RQ}$ is a path from $P$ to $Q$. Hence, $\mathbb{R}^2 -A$ is path connected.

Is my solution correct? Is my presentation and symbols correct?

asked Jul 31 '21 at 20:09

2

Duplicate here Indeed your proof is the same as Brian's one there. – Arctic Char Jul 31 '21 at 20:13
It's short and intuitive. It looks good to me. – John Douma Jul 31 '21 at 20:14

Show that if $A$ is countable subset of $\mathbb{R}^2$, then $\mathbb{R}^2 -A$ is path connected.

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