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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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    It is not true that the empty set must be in the basis.2012-11-16
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    Every open set must be a union of basis elements, so doesn't that imply that the empty set must be in the basis?2012-11-16
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    $\varnothing$ is the union of the empty set of elements of the base. That is, if $\mathscr{B}$ is the base, $\varnothing$ is one of the subsets of $\mathscr{B}$, and the open set $\varnothing=\bigcup\varnothing$.2012-11-16
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    Oh, okay, that makes sense. Thank you very much!2012-11-16

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