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Let $d$ be a metric on a set $X$, and let $ B=\{B(p,e) = \{y\in X \mid d(p, y)<\epsilon \}\text{for every $p\in X$ and every $\epsilon>0$}\} $ For $B$ to be the basis of a topology on $X$, then $\emptyset\in B$, but I don't see how this can be guaranteed since $\epsilon>0$.

Thanks so much!

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    Oh, okay, that makes sense. Thank you very much!2012-11-16

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The original answer is wrong, as you can see in the comments to it and to the question. Like the commenters said, you don't need the open set to be in the basis.

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    I ran out of space in the previous comment. Then the author proceeds by saying that for any $x$ in the empty set and any \delta >0, $(x-\delta,x+\delta) \subset \emptyset$. I see the bigger problem: I said that because the hypothesis is false, then the conclusion is true. And of course, that's nonsense. Sorry for the misleading answer and for taking the discussion completely off track. But I learned a valuable lesson out of it, so I guess it's not that bad. P.S. I see how my answer is a little different from what that book said. Sorry again for turning this thread into a huge mess.2012-11-16