224 METHODS OF MATHEMATICAL PROOF, PART II Chapter 6
interval is bounded on that interval" (which has as a corollary the result
that a continuous function on a closed and bounded interval attains an
absolute maximum and minimum value on that interval), and "a continuous
function on a closed and bounded interval is unvormly continuous on that
interval."
TRANSFINITE PROCESSESThere exists, among the axioms of set theory that constitute what is usually
called the foundations of modern mathematics, a collection of equivalent
statements that, because of their logical equivalence, constitute a single
axiom of set theory. This axiom, known in its various equivalent forms
as the axiom of choice, the principle of transfinite induction, the Hausdorf
maximal principle, Tukey's lemma, and perhaps most commonly, Zorn's
lemma, is widely used in existence proofs, where the goal is to prove the
existence of a "maximal" structure of some kind. Students are likely to
encounter applications of this axiom for the first time at the senior or begin-
ning graduate level. At a rather advanced stage of an introductory topology
course, Zorn's lemma is employed in the proof of the famous theorem of
Tychonoff, concerning the product of compact spaces. In abstract algebra
a Zorn's lemma argument is used to show that every field has an algebra-
ically closed extension field, while in linear algebra, the existence of a Hamel
basis for any vector space is proved by using the same general approach.
We note for your information the statement of Zorn's lemma: "A partially
ordered set S having the property that every chain in S is bounded above
in S has at least one maximal element." We mention also that this text
provides an introduction to partially ordered sets in Article 7.4 and a brief
consideration of the axiom of choice at the conclusion of Article 8.4.
Those of you who are motivated by the preceding discussion to pursue
at this stage one or more of the topics alluded to should consult, as appro-
priate, any of a number of introductory texts in abstract algebra, topology,
and mathematical analysis.