If a collection of closed sets of arbitrary cardinality in a metric space has empty intersection, does some countable subcollection?

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.


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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General Topology

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