Question

What are the countable dense subsets of real numbers?

Answer 1

This is a question from Rudin's Principles. Chapter 2, question 22.

The question reads: "A metric space is called separableseparable if it contains a countable dense subset. Show that RkRk is separable. Hint: Consider the set of points which have only rational coordinates."

The answer starts with: "We need to show that every non-empty open subset EE of RkRk contains a point with all coordinates rational." and then does just that, but I'm not sure how that addresses what the question is asking.

I know the definition of a dense subset. For a metric space XX and E⊂XE⊂X, EE is dense in XX if every point of XX is a point of EE or a limit point of EE (or both).

And I know that saying a set is countable means that the set has the same cardinality as the natural numbers or in other words could be put into one-to-one correspondence with the naturals.

Combining the two definitions to get definition of a countable dense subset is pretty straightforward.

And I understand that the rationals are countable. I also understand that the rationals are dense in RRwhich implies that QkQk is dense in RkRk.

But I don't know how showing that "every non-empty open subset EE of RkRk contains a point with all coordinates rational" proves that RkRk has a countable dense subset.