The statement included the word 'finitely many' thus excluding even countable union. At which point, the cardinality of the circle stops mattering I think. And

The statement included the word 'finitely many' thus excluding even countable union. At which point, the cardinality of the circle stops mattering I think. And at any rate, just... come on, we are obviously talking about a disk of radius r > 0 in the plane of real numbers using the euclidean metric.