Can't figure this 2 part question out - I think the first part involves using open balls but I'm not sure. It's straightforward to prove this for $\mathbb{R}^1$ with disjoint segments but am a little lost as to what to do when it comes to $\mathbb{R}^n$.

(1) Prove that every open set in $\mathbb{R}^n$ is the union of an at most countable collection of disjoint segments. Hint: You need to replace disjoint segments with the appropriate objects (which I'm thinking are open balls).

(2) Generalize the statement to a separable topological space. Hint: you may need some extra assumption.

Any help would be much appreciated! The more detail, the better! Thanks!

  • 3
    $\begingroup$ What exactly is a segment? $\endgroup$ – arkeet Dec 14 '12 at 7:12

Yes, open balls work here; just cover every point in the open set with an open ball; then because separable metric spaces are Lindelof, you can take a countable subcover.

  • $\begingroup$ Disjoint?${}{}$ $\endgroup$ – tomasz Feb 5 '14 at 22:45

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