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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?

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    If by lindelof property, you mean wether it is a lindelof space, then the answer is trivially yes. Compactness implies lindelof.2012-12-23
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    oh, yes, I have to sleep, so sorry.2012-12-23
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    Not 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

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