5.2. Compact and Perfect Sets

Examples 5.2.7(a):

Consider the collection of sets (0, 1/j) for all j > 0. What is the intersection of all of these sets ?
The intersection of all intervals (0, 1/j) is empty. To see this, take any real number x. If x 0 it is not in any of the intervals (0, 1/j), and hence not in their intersection. If x > 0, then there exists an integer N such that 0 < 1 / N < x. But then x is not in the set (0, 1 / N) and therefore x is not in the intersection. Therefore, the intersection is empty.

Note that this is an intersection of 'nested' sets, that is sets that are decreasing: every 'next' set is a subset of its predecessor.

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