Basic Analysis – Gently Done Topological Vector Spaces

28 Basic Analysis
Proposition 3.7 Suppose that F is any collection of subsets of a given set X
satisfyingthefip. ThenthereisamaximalcollectionDcontaining Fandsatisfying
the fip, i.e., if F ⊆ F ′ and if F ′ satisfies the fip, then F ′ ⊆ D. Furthermore,
(i) if A 1 ,...,A n ∈ D, then A 1 ∩···∩A n ∈ D, and
(ii) if A is any subset of X such that A∩D �= ∅ for all D ∈ D, then A ∈ D.
Proof As might be expected, we shall use Zorn’s lemma. Let C denote the
collection of those families of subsets of X which contain F and satisfy the fip.
Then F ∈ C, so C is not empty. Evidently, C is ordered by set-theoretic inclusion.
Suppose that Φ is a totally ordered set of families in C. Let A = �
S. Then
F ⊆ A, since F ⊆ S, for all S ∈ Φ. We shall show that A satisfies the fip. To
see this, let S1 ,...,S n ∈ A. Then each Si is an element of some family Si that
belongs to Φ. But Φ is totally ordered and so there is i0 such that Si ⊆ S for all i0
1 ≤ i ≤ n. Hence S1 ,...,S n ∈ Si0 and so S1 ∩···∩S n �= ∅ since S satisfies the i0
fip. It follows that A is an upper bound for Φ in C. Hence, by Zorn’s lemma, C
contains a maximal element, D, say.
(i) Now suppose that A 1 ,...,A n ∈ D and let B = A 1 ∩···∩A n . Let D ′ = D∪{B}.
Then any finite intersection of members of D ′ is equal to a finite intersection of
members of D. Thus D ′ satisfies the fip. Clearly, F ⊆ D ′ , and so, by maximality,
we deduce that D ′ = D. Thus B ∈ D.
(ii) Suppose that A ⊆ X and that A∩D �= ∅ for every D ∈ D. Let D ′ = D∪{A},
and let D 1 ,...,D m ∈ D ′ . If D i ∈ D, for all 1 ≤ i ≤ m, then D 1 ∩···∩D m �= ∅
since D satisfies the fip. If some D i = A and some D j �= A, then D 1 ∩···∩D m
has the form D 1 ∩···∩D k ∩A with D 1 ,...,D k ∈ D. By (i), D 1 ∩···∩D k ∈ D
and so, by hypothesis, A∩(D 1 ∩...D k ) �= ∅. Hence D ′ satisfies the fip and, again
by maximality, we have D ′ = D and thus A ∈ D.
We are now ready to prove Tychonov’s theorem which states that the product
of compact topological spaces is compact with respect to the product topology. In
fact we shall present three proofs. The first is based on the previous proposition,
the second (due to P. Chernoff) uses the idea of partial cluster points together
with Zorn’s lemma, and the third uses universal nets.
Department of Mathematics King’s College, London
S∈Φ