Let $(X,d)$ be a metric space. How to prove that every lindelöf metric space is separable?
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Hint: for each $n \in \mathbb{N}$, the cover $A_n := \{B_{\frac{1}{n}}(x) : x \in X\}$ of $X$ has a countable subcover, $$ \mathcal{B}_n = \{B_{\frac{1}{n}}(x^{(n)}_j)\}_{j \geq 1} \subseteq A_n. $$
In particular, each point of $x$ is in some ball of $B_n$. That is, each point of $x$ is at distance less than $\frac{1}{n}$ from some $x_j^{(n)}$. What does this tell you about $\bigcup_{n \in \mathbb{N}} \{x^{(n)}_j\}_{j \geq 1}$?
qualcuno
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1@Neilhawking what have you tried? – qualcuno Nov 08 '18 at 04:29
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1Well, if you aren't willing to make the effort to write down your ideas, you shouldn't expect people to help you. Maybe try to work out the original question from the hint, and take it from there. – qualcuno Nov 08 '18 at 04:35
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Done It is okay there was a missed line. Thanks Merci Danke – Neil hawking Nov 08 '18 at 04:44