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I have a set exercise which says:

Prove that in $\mathbb{R}^N$ with the Euclidean metric any collection of disjoint open sets is at most countable. Is this true for any arbitrary metric space?

Now I feel like I can do this question but I don't fully understand what it is asking.

Specifically: I don't understand the term "at most countable".

If someone could perhaps rephrase the question and give a hint (but NOT the answer) that would be appreciated. Thanks.

Ted
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user148533
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    I accidentally posted the incomplete question, have updated now. – user148533 Oct 19 '14 at 20:23
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    "at most countable" means that it is either finite or countably infinite. Or put another way, it may be infinite, but it is not uncountably infinite. Do you know about countably infinite and uncountably infinite sets? If not, now would be a good time to brush up. – MJD Oct 19 '14 at 20:23

1 Answers1

2

Hints:

  1. Every open set must contain a rational point (with all coordinate rational).
  2. Take a discrete metric space on some point set.
Berci
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