complement of compact set is connected Let A be a compact subset of R, the real numbers. Prove that the complement of A in the complex numbers C is connected. 
My thoughts: If A is compact then it is contained in a finite union. So if it's complement in C was disconnected it would imply C was disconnected-contradicton

My thoughts:
Whenever possible, I prefer dealing with path-connected spaces to connected spaces, because I can more easily visualize it.  If $A \subset \mathbb{R}$ is compact, then there's an $R > 0$ such that $A \subset [-R, R]$.  Now if we have two points $z, w \in \mathbb{C} \setminus A$, then we have a couple of easy cases:


*

*If only one of the two points happens to lie on $\mathbb{R}$ (say it's z), we can move vertically from $z$ to $\Re z + \Im W$ and then move along that line to $w$.

*If they happen to have the same nonzero imaginary part, then just connect them linearly.

*If they happen to have differing nonzero imaginary parts, just move horizontally (left or right) until you're passed $[-R,R]$, move vertically to the right half plane, and then you're free to move linearly.

*If they both happen to lie on $\mathbb{R}$, move vertically from one point into a half plane, move horizontally until you have the appropriate real coordinate, and then move vertically again.
