The union of open balls.

Question

Question

Show that every open subset of a metric space can be expressed as a union of open balls.

So far I have the following:

"Let $U \subseteq X$. For each $a \in U$, choose $r_a > 0$ such that $B(a, r_a) \subseteq U$."

I'm just not sure what the next step to show that $\bigcup_{a \in U}B(a, r_a) = U$.

Well, you've got two sets that you want to show are equal, so try showing that each is contained in the other. — Tara B, Apr 23 '12 at 11:04

score 8 · Accepted Answer · edited Apr 23 '12 at 12:10

8

$\bigcup_{a \in U}B(a, r_a) \subseteq U$, because it is, by definition, an union of subsets $B(a, r_a)$ of $U$.

$U \subseteq \bigcup_{a \in U}B(a, r_a)$, because each $a \in U$ is also contained in a right-hand side.

edited Apr 23 '12 at 12:10

answered Apr 23 '12 at 11:04

penartur

1 Answers1