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Let $(X,d)$ be a nonseparable metric space. It follows from here that there exists an uncountable subset $N$ of $X$ and $r>0$ such that for all $x\neq y$ in $N$ we have $d(x,y)>r$.

Is it possible to show that such $N$ must have cardinality greater than or equal to $\mathbb R$?

Thanks a lot for your help.

José Carlos Santos
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Alphie
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1 Answers1

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It depends upon whether or not you accept the continuum hypothesis. If you do, then the statement is automatically true, because then the cardinal of any uncountable is greater than or equal to the cardinal of $\Bbb R$.

Otherwise, take an uncountable set $X$ whose cardinal is smaller than that of $\Bbb R$, consider on it the discrete metric and take $N=X$. Then, if $x,y\in N$ and $x\ne y$, $d(x,y)=1>\frac12$.

José Carlos Santos
  • 427,504
  • Thank you for your answer. That's embarrassing as I cannot commit on the continum hypothesis being true or false. Can I at least say that the power set $\mathcal{P} (N)$ has cardinality strictly greater than $c$? – Alphie May 01 '21 at 17:20
  • No. They have the same cardinal. – José Carlos Santos May 01 '21 at 17:21
  • Ok I got this question from here : https://math.stackexchange.com/q/4123497/522332 but I must be missing the point of the argument. Would you mind to have a look? Thanks a lot. – Alphie May 01 '21 at 17:39