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So I have a metric space $(X,d)$ and a closed subset $P\subset X$, where $P$ has no isolated points, and I want to build a sequence of nested balls in the following way:

  • Need to ensure that there are two balls $B_1$ and $B_2$ totally contained in $P$, with $B_1 \cap B_2=\emptyset$.

So it is evident that we can take a radius and two points in $P$ so that that $B_1 \cap B_2=\emptyset$. But how do we ensure that $B_1$ and $B_2$ are totally contained in $P$? I'm interested in the formal construction of such two (closed) balls. Then I know how to construct the rest of the sequence.

Would appreciate some help.

sequence
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1 Answers1

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The cantor set is a counter example, as a totally disconnected compact set with no isolated points, considered as a subset of $\mathbb{R} $ with the usual metric.

Noah Riggenbach
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  • So I probably don't even need that ball to be contained in $P$. But then I don't see how such a sequence would necessarily converge in $P$. Edit: No, actually it will, because we are always picking points in $P$ for each subsequent ball. – sequence Oct 19 '17 at 22:52
  • @sequence I don't understand, I think there is something missing in your question. – Noah Riggenbach Oct 19 '17 at 22:56
  • Sorry, this is a subquestion of my main proof at https://math.stackexchange.com/questions/2480637/cardinality-of-perfect-set-in-compete-metric-space-is-at-least-continuum

    (This is not a duplicate, as I'll soon edit that main post, I just needed to resolve two main issues there).

     – sequence Oct 19 '17 at 23:00