Let X be a connected metric space, let a, b be distinct points of X and let r > 0. Is there a collection {B_i} of finitely many open balls of radius r such that their union is connected and contains a and b.(adsbygoogle = window.adsbygoogle || []).push({});

I was trying to prove this by contradiction, but couldn't derive a contradiction. I can't think of any counterexamples either. Does anyone know whether this statements is true or false?

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# Connected Metric Space Question

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