- #1
ArcanaNoir
- 779
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Let {Q} be a collection of cubes covering a set E in R^n. Prove that there is a countable sub collection {Q}' of these cubes which covers E and [tex] \cup {\frac{1}{2}Q} \subseteq \cup {Q}' [/tex], and the number of cubes in the subcollection containing any given point of E is less than something depending only on the dimension.
Oh and the sup of the side lengths of the cubes is bounded, but E is not bounded.
[tex]\frac{1}{2}Q [/tex] means the cube inside Q with side length 1/2 of the side length of Q.
In other words, the countable subcovering needs to cover the middle sections of the original cover.
I really like this proof Real Analysis - Emmanuele DiBenedetto - Google Books
But in this proof the set E is the centers of the cubes, so we only end up covering centers, not middle halves, and also E is bounded in that proof. My professor said I could modify this proof but I don't see how. *sad face* He also said the first half of my most recent attempt at this proof was badly written and the second half was worse. :(:(
The idea for dealing with the unbounded set E is to take "layers" of cubes which are about the same size. So something like sup{S(Q)}=R where S(Q) is the side length of Q and then [tex](1-\epsilon)^kR< S(Q)\le R[/tex] for the first layer, then take a maximal set of points in E which are some specific distance apart (this is a piece I messed up on, no matter what distance I separate my points by there is always some flaw in my choice). Then take the cubes containing those points and discard some cubes (which ones?). Rinse and repeat.
I tried doing some optimal arrangement of separated points around a specific point for the bound, but I was told I did that wrong too :( (that was the "worse" part apparently.)
Oh and the sup of the side lengths of the cubes is bounded, but E is not bounded.
[tex]\frac{1}{2}Q [/tex] means the cube inside Q with side length 1/2 of the side length of Q.
In other words, the countable subcovering needs to cover the middle sections of the original cover.
I really like this proof Real Analysis - Emmanuele DiBenedetto - Google Books
But in this proof the set E is the centers of the cubes, so we only end up covering centers, not middle halves, and also E is bounded in that proof. My professor said I could modify this proof but I don't see how. *sad face* He also said the first half of my most recent attempt at this proof was badly written and the second half was worse. :(:(
The idea for dealing with the unbounded set E is to take "layers" of cubes which are about the same size. So something like sup{S(Q)}=R where S(Q) is the side length of Q and then [tex](1-\epsilon)^kR< S(Q)\le R[/tex] for the first layer, then take a maximal set of points in E which are some specific distance apart (this is a piece I messed up on, no matter what distance I separate my points by there is always some flaw in my choice). Then take the cubes containing those points and discard some cubes (which ones?). Rinse and repeat.
I tried doing some optimal arrangement of separated points around a specific point for the bound, but I was told I did that wrong too :( (that was the "worse" part apparently.)
