From Osher Doctorow
Edwin Hewitt and Karl Stromberg of U. Oregon in Real and Abstract
Analysis, Springer-Verlag: N.Y. 1965, a classic reference and textbook
in graduate real analysis, define the "finite intersection property"
as: each finite subfamily of sets has nonvoid intersection. They
prove:
Theorem 1. A topological space X is compact iff each family of closed
subsets of X with the finite intersection property has nonvoid
intersection.
Theorem 2. Let X be a metric space. Then the following three
assertions are pairwise equivalent. (i) X is compact, (ii) X is Brechet
compact, (iii) X is sequentially compact.
These are proved on page 53 of Hewitt and Stromberg.
That the finite intersection property is a key to compactness also may
be suspected from the definition of a topological space as compact if
every open cover of the space admits a finite subcover, which has a
similar to Theorem 1.
There are also some very important fundamental relationships in
mathematics (including especially set theory) and logic which relate
directly or indirectly to the finite intersection property. One is
"finite character" (a set A is in a family of sets F iff each finite
subset of A is in F) which relates to Tukey's Lemma:
Tukey's Lemma (Hewitt and Stromberg p. 14). Every nonvoid family of
finite character has a maximal number.
Furthermore, we have (p. 14 of Hewitt and Stromberg):
Theorem. The following finite propositions are pairwise equivalent:
(i) the axiom of choice, (ii) Tukey's Lemma, (iii) the Hausdorff
maximality principle (every nonvoid partially ordered set contains a
maximal chain), (iv) Zorn's lemma (ditto for each nonvoid partially
ordered set in which each chain has an upper bound), (v) the
well-ordering theorem (every set S can be well-ordered, i.e. there is a
well-ordering < = on S).
Osher Doctorow
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