Plane minus a countable set is path connected
= Plane minus a countable set is path connected
For a <countable set> $T\subset\mathbb R^2$, join two points outside $T$ through an auxiliary point avoiding the countably many lines through either endpoint and a point of $T$. Such an auxiliary point exists: choose a line with direction outside the countable list of forbidden directions, then avoid its countably many intersections with those lines. The two segments avoid $T$, giving a <continuous path>.