$X$ 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
General Topology – Metrizable Lindelöf Space has a Countable Basis
general-topology
Best Answer
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 $\mathscr{B}=\bigcup_{n\in\Bbb Z^+}\mathscr{B}_n$.