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In this question I claim that every nested sequence of bounded closed subsets of a metric space has nonempty intersection if and only if the space has the Heine-Borel property. However, there's something that can throw a wrench in the proof: what if it is possible for there to be an uncountable collection of closed subsets with empty intersection such that every countable subcollection has nonempty intersection?

Is this possible in a metric space?

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Let $X$ be an uncountable set endowed with the discrete metric. Then the family $\{X\setminus\{x\}\,|\,x\in X\}$ is an uncountable family of closed subsets with empty intersection. But no countable subfamily has that property.

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  • $\begingroup$ Of course. Thank you. $\endgroup$ – Matt Samuel Feb 9 at 22:46
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The property that "if a family of closed sets has an empty intersection, then there is a countable subfamily with empty intersection", has a name. It's called Lindelöf. In a metric space this is equivalent to having a countable dense subset (separable), and many other such countability properties.

Hence Santos' example was the standard example of a non-separable metric space.

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