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Let $O$ be open in $\mathbb{R}^n$. Show that O is the union of a sequence of compact sets $(C_n)$ such that $C_n\subset (C_{n+1})^o$.

The answer seems pretty simple when O is bounded, since $cl(O)$ will be compact, and I can construct the sequence from there by the Lebesgue epsilon property. But how to do the non bounded case? Would it work if I cut the set into bounded pieces? (Guess not...)

MelaniesWoes
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1 Answers1

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$$\mathbb R^n\cong (0,1)^n$$

You already know how to solve the problem for bounded open sets (e.g. open subsets of $(0,1)^n$) but compactness is a topological property (preserved by homeomorphism) so take your problem from lhs to rhs, solve it, then take it back.

A simple example of the solution for $O=(0,\infty)$ is to take $C_n=[1/n,n].$

Dan Robertson
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  • Lovely! The compacta are the inverse images of the distance function. Thanks. Voted up – MelaniesWoes Dec 26 '17 at 22:37
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    @MelaniesWoes I’m not sure I understand what you mean. The example I give is to illustrate how the problem may be solved for non bounded sets. If one were to solve the problem for $(0,\infty)$ translated into a problem within $(0,1)$ then the preimage of the solution under that map would likely not be as I have in the example. – Dan Robertson Dec 26 '17 at 22:45