I know that uncountably product of $\{0,2\}$ is not-metrizable, separable, compact, Hausdorff and not second countable. But what we can say about Lindelöf property? I think it is not Lindelöf, but how can I show it? Could you give me any idea?
uncountable product of two-point discrete space
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general-topology
compactness
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0Not all uncountable products of a 2-point space are separable (you need at most continuum many copies). All such spaces are ccc though. – 2012-12-23
1 Answers
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It is obviously Lindelöf (every open cover has a countable subcover) as it is already compact (every open cover has a finite subcover). It is not hereditarily Lindelöf, nor hereditarily separable, as the subspace of all points with exactly one coordinate equal to 2 (and all others 0) is discrete and uncountable, and so not Lindelöf and not separable.