Let $(X,d)$ be a metric space. Suppose that $(X,d)$ has the property that every family of non-empty, pairwise disjoint open sets in $X$ is countable. Is $X$ second countable then?
A characterization of second countability
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general-topology
metric-spaces
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0Look at http://math.stackexchange.com/questions/90427/countable-chain-condition-for-separable-spaces – 2012-06-08
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0In the metrizable space $X$, the following conditions are equivalent, cf. General Topology by Engelking Page 255: > $X$ is seperable > $X$ is second countable > $X$ is lindelof > $X$ has countable extent > $X$ is star countable > $X$ is CCC – 2013-05-20