I am reading a paper and they make a claim, which I was having trouble following.

Suppose I have a bounded set $B$ inside $\mathbb{R}^n$. Let $U_1, ..., U_M$ be an open cover of $B$. Then is it always possible to find closed sets $C_1, ..., C_M$ such that each $C_i$'s are disjoint, $C_i$ is contained in $U_i$ and $C_i$ is bounded by hyperplanes parallel to the co-ordinate hyperplanes?

It appears that this is what they are doing, and I wasn't sure how this was possible. (I was also wondering that even though they say disjoint maybe they are allowing intersections on the boundary? but I wasn't sure...) Any comments/explanations are appreciated. Thank you very much.

  • $\begingroup$ Just to make the problem interesting, require the C's to be not empty. Pick m unique unique points, one from each U. $\endgroup$ – William Elliot Nov 17 '17 at 20:04
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    $\begingroup$ $(0,1)$ covers $(0,1)$. No such closed $C$. What am I misunderstanding? $\endgroup$ – jdods Nov 18 '17 at 4:15
  • $\begingroup$ @jdods hmmm. It looks like I am missing something... I will go back to the paper and fix this question. Thank you. $\endgroup$ – Johnny T. Nov 19 '17 at 16:43
  • $\begingroup$ @JohnnyT. Well, the statement of your question (aside from the title) doesn't make it clear that the $C_i$ are to form a cover of $B$. So maybe they are not meant to form a closed cover? Or maybe there are further details that need to be included in the question... $\endgroup$ – jdods Nov 19 '17 at 18:02

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