Chauvenet's treatise on elementary geometry by W. E. Byerly

By W. E. Byerly

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Be a collection of closed subsets of S satisfying the finite intersection hypothesis. Then there exists a collection 5' of subsets of S such that 5' contains 5, 5' satisfies the finite intersection hypothesis, and 5' is not a proper subcollection of any other collection of sets having the first two properties. Proof: Let n = {5,} be the family of all collections of subsets, not necessarily closed, of S, such that 5 fJ satisfies the finite intersection hypothesis. Partially order n by defining 5"' < 5, to mean that every set in 5"' is a set in 5, but not conversely.

One maximal subset M is the set of all points (n, 5), n > 0. Another is the set (n, y), where y = 5 if n is a prime andy = 3 otherwise. For a more complete discussion of the various forms of the axiom of choice, the reader is referred to Wilder [43], or to Fraenkel-Bar Hillel [8(a)]. 1-10 The Tychonoff theorem. A principal justification for adopting the Tychonoff topology in product spaces is the following result. THEOREM 1-28. If {S.. } is any collection of compact spaces, indexed by an index set A, then the product space IP AS..

The word "compact" has been defined in so many (related) ways that one must be quite careful in reading the literature. For a long time, a space was said to be compact if it were what we have called countably compact. And a subset X of a space S was said to be compact if every infinite subset of X had a limit point in S. In metric spaces, which were then the most widely studied, our compactness for spaces was proved as a theorem, and our compactness for subsets was shown to be equivalent to the old compactness for subsets plus closure.

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