# Thread: Connectedness of a subset of R^2

1. ## Connectedness of a subset of R^2

Hi I need help with this problem. This is not for a grade. Please give a detailed proof, this is just to help me study.

1) Let C be a countable subset of the plane $\displaystyle \mathbb{R}^2$. Prove the complement of C in $\displaystyle \mathbb{R}^2$ is connected.

2. Originally Posted by ElieWiesel
Hi I need help with this problem. This is not for a grade. Please give a detailed proof, this is just to help me study.

1) Let C be a countable subset of the plane $\displaystyle \mathbb{R}^2$. Prove the complement of C in $\displaystyle \mathbb{R}^2$ is connected.
Every path-connected space is connected. So I think it is easier to show that it is path-connected. The proof for path-connectedness of the above problem can be found here.

3. Originally Posted by aliceinwonderland
Every path-connected space is connected. So I think it is easier to show that it is path-connected. The proof for path-connectedness of the above problem can be found here.
There's a slightly simpler argument than the one presented. Given any two points x and y in R^2 - C, there are uncountably many pairwise disjoint (except at the endpoints) paths from x to y. (E.g., take any point z on a given line perpendicular to that between x and y, and then draw the path from x to z to y.) Since there are only countably many points in C, at least one of these paths (actually, uncountably many) must be free of any of these points. Hence R^2 - C is path-connected.

4. Awesome, thanks! I actually came up with something similar. Its nice to know I was on the right "path"...get it...get it. "Path connected" oh man i crack myself up.

Thanks again.