I am having a little trouble with path connectedness.

I am trying to determine whether or not the set R^2\{(0,0)} is path connected.

My first thought was that since R^2\{(0,0)} is a subset of R^2 and R^2, being the cartesian product RxR, is path connected, it follows that R^2\{(0,0)} cannot be both open and closed and thus cannot be path connected.

However, when I imagine it, it seems that given any two non-zero points in R^2, I can connect them by the straight-line path between them (if it does not go through the origin) or I can choose a third point with which to form a broken line connection, perfectly satisfactory as a path.

So that would imply that R^2\{(0,0)} is in fact path connected.

So I think I need clarification of the following iff statement:

A space X is connected iff the only subsets of X that are both open and closed in X are the empty set and X itself.

So, do I understand correctly that this statement says that every path connected set is open and closed? And if so, does this statement not say that no subset of a path connected set can be path connected? It doesn't seem to make sense. What does this statment mean for my space R^2\{(0,0)}?

Any help on the statement as well as the properties (open, closed, connected, path connected) of R^2\{(0,0)} would be greatly appreciated.