Failure of Baire Category Theorem if collection of sets is not closed or not countable.

Question

The Baire Category Theorem implies that $\mathbb{R}^n$ cannot be written as the union of a countable collection of closed subsets having empty interior.

I was wondering how it can be showed that this fails if the sets are not specified to be closed, or the collection is not required to be countable.

Help is appreciated. Thanks

Answer 1

If sets are not closed, you can get by with two sets: the irrationals and the rationals.

If the collection is not countable: all singletons.