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