If $C⊆ℝ$ is the Cantor set, what is the rank of the (necessarily free) fundamental group $π_1(ℝ^2 - C×\{0\})$?
Since the complement of the Cantor set is open, and an open set in $ℝ$ is always a union of at most countably many disjoint open intervals, the number of "gaps" between the points of the Cantor set (through which we may braid loops in the plane) is countable. This leads me to believe that there is a countable set of generators for the fundamental group in question, if indeed every loop may be deformed to such a well-behaved finite braiding. However, I don't know how to proceed.