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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?

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The issue is that if you take finite number of points $\{x_1,\dots,x_k\dots,x_K\}$ and write the metric space as union of these points. Each singleton is not nowhere dense anymore because each singleton itself is a sphere so it is dense in itself.

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