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 . Prove the complement of C in is connected.
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.