# A metrizable Lindelöf space has a countable basis

$X$ is called metrizable Lindelöf space if $X$ is a metrizable space and every open covering of $X$ contains a countable subcovering. Would you help me prove that $X$ has a countable basis? Thanks

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This question from Munkres book exercise 30 –  beginner Nov 10 '12 at 6:26
Note that every compact space is Lindelöf, so this result implies the one in your other question. –  Brian M. Scott Nov 10 '12 at 6:27

HINT: Essentially the same hint that I gave for this question works here. For each positive integer $n$ let $\mathscr{U}_n=\left\{B\left(x,\frac1n\right):x\in X\right\}$; this is an open cover of $X$, so it has a countable subcover $\mathscr{B}_n$. Consider $\mathscr{B}=\bigcup_{n\in\Bbb Z^+}\mathscr{B}_n$.

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In the metrizable space $X$, the following conditions are equivalent, cf. General Topology by Engelking Page 255:

$X$ is separable

$X$ is second countable

$X$ is Lindelöf

$X$ has countable extent

$X$ is star countable

$X$ is CCC

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