Let $A$ be a countable subset of $\mathbb{R}^n$, where $n > 1$.

Prove that $\mathbb{R}^n \setminus A$ is path-connected.

Analogy of R^2. Notice that n = 1 does't work.

PICTURE ATTACHED!

Please Help!!!

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- October 16th 2012, 02:28 AMvercammenPath connectedness in R^n
Let $A$ be a countable subset of $\mathbb{R}^n$, where $n > 1$.

Prove that $\mathbb{R}^n \setminus A$ is path-connected.

Analogy of R^2. Notice that n = 1 does't work.

PICTURE ATTACHED!

Please Help!!! - October 16th 2012, 08:14 AMxxp9Re: Path connectedness in R^n
for any two points x, y not in A, pick any single parameter family of curves that connects x and y, having only x, y as their common points. There must be one curve c in the family such that c and A have no common points. Otherwise, suppose each curve intersects with A, A will have so many points that are not countable.

- October 16th 2012, 09:05 AMvercammenRe: Path connectedness in R^n
thank you!

- October 16th 2012, 09:21 AMjohnsomeoneRe: Path connectedness in R^n
Here's an explicit family of paths serving the purpose of xxp9's post:

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- October 16th 2012, 11:34 AMvercammenRe: Path connectedness in R^n
Thank you!