Proposition 2.4.7 * is a metric space with metric . is a sequence of descending, non-empty, compact sets.
Then for in the Hausdorff sense.
In that sense, one must show that
(1) and (2)
Prof Edgar shows (1) clearly, but I have difficulty with his demonstration of (2). I shall try to restate it:
By the neighborhood of , he means:
. is an open set, he states, which is straightforward.
He then states that the family
is an open cover of . Herein is the complement of . It seems to me that the family described covers not only but the entire metric space , all points of the intersection and all points "outside" the intersection . So, yes, it does cover . The author then continues that there is therefore a finite subcover of , since A_1 is compact by hypothesis. He states that:
for all , where M is some positive integer is a finite sub-cover. The author concludes with
Therefore, , and (2), above, is satisfied.
My difficulty: The expression for the finite sub-cover does not seem finite to me because it is a family for all n > M.
Can anyone explain?
*"Measure, Topology, and Fractal Geometry", by Gerald A. Edgar (Springer Verlag, New York, 2000) p.68