Assume that \(\mathbb{R}\) is uncountable. Show that if \(A\) is a countable subset of \(\mathbb{R}^2\), then \(\mathbb{R}^2 - A\) is path connected. [Hint: How many lines are there passing through a given point of \(\mathbb{R}^2\)?]
Solution
Let \(p,q\in\mathbb R^2-A\). There are uncountably many lines through \(p\), while only countably many of them contain a point of \(A\); also at most one contains \(q\). Choose a line \(L\) through \(p\) that contains neither \(q\) nor any point of \(A\).
For each \(a\in A\), the line through \(q\) and \(a\) is different from \(L\), because \(q\notin L\), so it meets \(L\) in at most one point. Let \(E\subseteq L\) be the countable set of these intersection points. Since \(L\) is uncountable, choose \(r\in L-E\).
The segment \([p,r]\) lies in \(L\) and therefore avoids \(A\). If \([q,r]\) met \(A\) at \(a\), then \(r\) would lie on the line through \(q\) and \(a\), contrary to the choice of \(r\). Thus the broken line \[ p\longrightarrow r\longrightarrow q \] is a path in \(\mathbb R^2-A\). Therefore \(\mathbb R^2-A\) is path connected.