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I began reading Rudin's Real and Complex Analysis, and I have a question about the following:

Rudin defines a topology $T$ in a set $X$ as the collection of subsets of $X$ such that

(i) empty set and $X$ are members of $T$ (ii) members of $T$ are closed under arbitrary unions (iii) members of $T$ are closed under finite intersections.

He then says that if $X$ is a metric space and $T$ is the collection of all subsets of $X$ which are the arbitrary unions of open balls, then $T$ is a topology in $X$.

What I do not understand is how is empty set the arbitrary union of open balls?

Can anyone explain this to me?

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    $\begingroup$ The empty union is a union. $\endgroup$
    – Thomas Andrews
    Jan 24, 2015 at 5:06
  • $\begingroup$ @ThomasAndrews what do you mean by the empty union? $\endgroup$
    – mononono
    Jan 24, 2015 at 5:06

1 Answer 1

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The empty set is the union of the empty family of open balls. More generally, if $\mathscr{B}$ is any family of sets, $x\in\bigcup\mathscr{B}$ if and only if there is a $B\in\mathscr{B}$ such that $x\in B$. If $\mathscr{B}=\varnothing$, then there is no $B\in\mathscr{B}$ at all, so it doesn’t matter what $x$ is: it can’t be in $\bigcup\mathscr{B}$. Thus, $\bigcup\mathscr{B}=\varnothing$ when $\mathscr{B}=\varnothing$.

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