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I'm in my first analysis class and I'm pretty confused about what direction to take this question. The definitions I'm working with are:

Open Base: $(U_{\alpha})$ is an open base of M if every open subset of $M$ can be written as the union of $U_{\alpha}$.

Dense: A set $A$ is dense in $M$ if if $\bar{A}=M$ $\iff$ $\forall x\in M, \forall\varepsilon>0,$ $B_{\varepsilon}(x)\cap A\not=\phi$

Separable: A metric space $M$ is separable if it contains a countable dense subset.

I got a hit that we should consider $\{x_n\}$ a countable dense subset in $M$ and the collection of balls with rational radii centered at $x_n$, and I how we could potentially use this to construct an open base... but not how to go from an arbitrary open base to separable.

Thanks for any help!

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Going from a countable base to separability is actually the easier direction. Let $\mathscr{B}=\{B_n:n\in\Bbb N\}$ be a base for $M$, and for each $n\in\Bbb N$ let $x_n\in B_n$. Let $D=\{x_n:n\in\Bbb N\}$; now use the fact that $\mathscr{B}$ is a base to show that $D$ is dense in $M$.

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  • $\begingroup$ Thanks! This is the first time I'm seeing these objects, so I really didn't know where to start. $\endgroup$ Commented Oct 22, 2020 at 18:49
  • $\begingroup$ @TannerFields: You’re welcome! $\endgroup$ Commented Oct 22, 2020 at 18:49

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