Interactive Real Analysis

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5.2. Compact and Perfect Sets

Examples 5.2.5(b):

Let S = [0, 1]. Define = { t R : | t - | < and S} for a fixed > 0. Is the collection of all { }, S, an open cover for S ? How many sets of type are actually needed to cover S ?
First, each set is an open set, because it is the same as an interval around of length 2 . Second, the union of all sets equals the open interval (- , 1 + ), so it contains the set S. Therefore, the collection { }, S is an open cover of S.

The collection { }, S consists of uncountable many sets. In order to cover S, however, we need only a finite subcollection for any given . To see this, fix an > 0. Then let N be the smallest integer greater than 1 / , and define

• = k * , k = 0, 1, 2, ... N
Then one can quickly check that the collection { }, k = 0, 1, 2, ..., N is a covering of S. That is, this new collection forms a finite subcover of S with respect to the original collection of sets.
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