I was asked to prove, every punctured open set in R^2 is path connected. My argument : take points x and y. let z be the point we've taken off from U (open). if x, y,z do not pass through a staright line, we have a segment between a and y. Now if the 3, i.e. x,y,z lie on a straight line...then pick another point ,say p, not lying on the staright line. we have segments joining x and p and p and y. hence, we've found a path between x and y ,as required. apparently, this is wrong...my prof told me not bring in straight lines anywhere! any hints?? thanks!