Question about infinite metric spaces and open sets

Question

To prove :Let M be an infinite metric space. Prove M contains an open set U s.t U and $U^c $ are infinite

I am thinking to use contrapositive technique (~q-> ~p)

The Contrapositive Version

If U and $U^c $ are not open and finite show M is a finite metric space ( Can’t assume closed sets. I am on section 4.2) I am using set theory and metric spaces by Kaplansky I can use everything up to 4.2

But if I decide the direct technique how do I argue with infinite subsets and metric spaces ?

I know that the union of open sets is opened Ref: https://math.stackexchange.com/a/2118888/748810

So by the Ref U is opened and infinite.( A wild guess. I hope I don’t choke)

Answer 1

Hints:

Consider two cases:

(I) The metric space is discrete. Then the claim is evident.

(II) The metric space is not discrete. Find an injective sequence that converges and consider the set of its terms with odd index together with its limit. Show that this set is closed (for example, show it is compact) and let $U$ be the complement of this set.