I am still having problems with some of the proofs of Baire's theorem.
In Introductory Real Analysis by Kolmogorov and Fomin
The statement of the theorem states that:
"A complete metric space R cannot be represented as the union of a countable number of nowhere dense sets."
I understand that a metric space with a finite number of points is complete.
So if we have a complete metric space with K number of points, why is this complete metric space not represented by the union of k singletons, X_1, X_2 ... ... X_k?