YYet another proof of Tychonoff's Theorem N. Noble Abstract
In 1971 I announced a “nice” proof of Tychonoff's Theorem, an immediate corollary of a resultconcerning closed projections combined with Mrówka's characterization of compactness: a space X iscompact if and only if for each space Y the projection π Y : X x Y Y is closed. I described the proof as“to appear” but to date it has not. In 2019 I published a generalization of the stronger closed projectionresult which yields a different look to the “nice” proof. Both versions are presented here.
0 Introduction
In Noble [1971] I suggested that “the proof of Tychonoff's theorem using Theorem 1.8 is also ofsome interest. Its use of the axiom of choice differs from that of other proofs in that it is restricted tostatements about the index set” and referred the reader to “A nice proof of Tychonoff's Theorem (toappear)”. That proof did not “appear”; the draft article was never finished and no longer exists. Myvague recollection of it is that it used the proof as a springboard for examining variations of the Axiomof Choice. All that remains, aside from that recollection, is a footnote in Halpern, [1972, p119] whichmentions, with regard to its relationship to some other conditions of the statement "Every infinite setalgebra has a non-principal prime ideal.", that “The referee has informed us that this proof wasdiscovered independently by Norman Noble.” . Fast forward: recently Matheron[2020] published three proofs of Tychonoff's Theorem, one ofwhich relies upon Mrówka's characterization of compactness (and triggered my recollection of thesubject of this note). Matheron's proof was extracted from a more general categorical result inClementino & Tholen [1996] , as was a similar proof given by Todd Trimble in Mathoverflow 26416[2016]. Another proof using Mrówka's characterization is contained in Jachymski [2005], where someof the other consequences of my Theorem 1.8 are rediscovered.
1 The “ nice ” proof
For π Y the projection from X x Y onto Y my 1971 proof exploits Mrówka's characterization ofcompactness in the form: for S a subset of X x Y, if π Y S is not closed then S is not closed. The proof isa transfinite induction which, for π Y S not closed, constructs a point x in clS \ S. The induction step is abarely deeper characterization of closed projections, stated next.
The projection π Y : X x Y Y is closed if and only if for each subset S of X x Y andeach point y of cl(π Y S) there exists a point x of Y such that (x,y) is in cl(S).
1 This title is chosen in fond memory of Ernie Michael who will not otherwise be mentioned in this note.2 P.O. Box 87, Florence, OR 97439; norm@metavideo,com
MSC: primary 54B10, 54D30 secondary 54C10,
Keywords:
Tychonoff's Theorem, closed projections.3 Halpern's article uses and generalizes a forcing technique credited to David Pincus. David was a friend and collegeclassmate of mine and, circa 1970, a colleague at the University of Washington where we frequently discussed suchmatters. I suspect he was the referee cited.4 It is presented in a fuller context in Clementino, Giuli & Tholen [2004].5 Due, per Engelking [1989, p133], to Kuratowski [1931], Bourbaki [1940], and Mrówka [1959] . 1 roof.
First suppose π Y is closed, S X x Y, and y cl(π Y S). Since π Y cl(S) is a closed set containingπ Y S, y is in π Y cl(S), that is, X x {y} meets cl(S). For any point x in that intersection, (x,y) is in cl(S).For the converse, if π Y is not close there exists a closed subset S of X x Y with a point y in cl(π Y S) \ π Y Sand for that y, no point (x,y) is in cl(S) = S. Notation.
For X = Π α κ X α , κ a cardinal, and β ≤ κ , set X β = Π α < β X α , so X β+1 = Π α ≤ β X α = X β x X β . Let π β be the projection from X β+1 along X β onto X β and for S a subset of X set S β = π β (S). (Noble [1971, Theorem 1.8) . Let X = Π α κ X α where κ is a limit ordinal. If for each βsatisfying 0 < β < κ the projection of X β+1 onto X β is closed, then for each β < κ the projection of X ontoX β is closed. Proof.
Suppose β < κ and S X are such that S β is not closed; by setting X β equal to X and, for α > β,re-indexing X α as X α - β , we may suppose β =
0. Choose a point x in cl(S ) \ S and for β > β in X β such that (x ,...,x β ) is in cl(S β+1 ). Notice that foreach β ≤ κ the point (x α : α < β ) is in the closure of S β, either by construction or, for β a limit ordinal,because verification that a basic open neighborhood of a point meets a subset of a product depends ononly finitely many coordinates. In particular, the point (x α : α < κ) is in cl(S). Since x is not in S thispoint is not in S, S is not closed, and therefore the projection of X onto each X β is closed. (Tyconoff's Theorem). Each product of compact spaces is compact. Proof.
Let X be the product of a collection {X α : 0 < α < κ } of compact spaces with X any space andκ any non-zero cardinal, and note that since the projection along each X α is closed, Theorem 1.2implies that the projection along X onto X is closed. Hence by Mrówka's Theorem X is compact.
2 The second proof
Recently I used Theorem 1.2 to prove a more powerful result, repeated below. (Noble [2019, Theorem 6.2]) . Let κ be an infinite cardinal, suppose R is a class of topological spaces closed under products offewer than κ factors, and define P and Q as follows:• P = {X: π Y : X x Y Y is closed for each Y in R }; and• Q = P ∩ R .If X = Π α A X α where each X α is in Q , then(a) if |A| < κ , X is in Q ; and(b) if |A| = κ , X is in P . Proof.
First note that the class Q is closed under finite products; indeed, for X P , and Y, Z R ,π YxZ : X x Y x Z Y x Z is closed, and for Y P , Z R, π Z : Y x Z Z and thus π: X x Y x Z Z is closed.Since this is true for each Z R , XxY is in P . Since X,Y Q implies also that X x Y is in R , X x Y is in Q . To complete the proof, note that X = X ≤ is in Q and suppose inductively that X < β is in Q for allβ < α. If α = β + ≤ α is in Q as a finite product of members. Otherwise eachπ: X ≤ α X < α is closed since X α P and X < α R , so, by Theorem 2.1, X is in P . If |α| < κ then, as aproduct of fewer than κ factors, X is also in R and thus Q .Theorem 2.1 provides a process, given a judicious choice of the class R , for generating producttheorems involving κ-fold products: Vaughan [1975, Theorem 2] provides several examples. Thelimits of that process are suggested by two standard arguments reviewed below.2 .2 Remark For spaces X and Y let π Y : X x Y Y be the projection and let κ be an infinite cardinal.(a) If X has a κ -fold open cover {U α : α κ} with no subcover of smaller cardinality and Y contains anincreasing collection {H α : α κ} of closed sets whose union is not closed, then π Y is not closed.(b) If X is [κ,∞]-compact (each open cover has a subcover of cardinality less than κ ) and in Y eachintersection of fewer than κ open sets is open, then π Y is closed.(c) If π Y is closed and Y is a product with κ nontrivial (not indiscrete) factors, then X is initiallyκ-compact (each κ-fold or smaller open cover has a finite subcover). Proof. (a). Set F α = ∩ β < α (X \ U β ); then S = U α < κ F α x H α is closed but π Y S = U α H α is not. (b). If S is a closed subset of X x Y and y is a point of Y not in π Y S, we can cover X x {y} withκ-many basic open subsets {U α x V α : α κ} which do not meet S, demonstrating that ∩ α V α is aneighborhood of y which does not meet π Y S. Hence π Y S is closed and therefore π Y is closed.(c). Suppose Y = Π α κ Y α where each Y α contains a non-empty proper closed subset C α and set H α =Π β ≤ α X β x Π α < β C β . Note that if λ ≤ κ is an infinite cardinal, {H α : α λ} is an increasing collection ofclosed sets whose union is not closed; thus by (a) and the fact that π Y is closed any infinite cardinal λsuch that X has a λ-fold open cover with no subcover of smaller cardinality must be greater than κ. Itfollows that X is initially κ-compact.In particular, if X is a Lindelöf space which is not compact then for any space Y the projection π Y on X x Y is closed if and only if Y is a P-space. Notice that a product of Lindelöf P-spaces with acountably infinite number of nontrivial factors is not a P-space, so 2.1(b) cannot be improved toconclude that X is in Q . Of course, our interest here is with the application of 2.1 to the simplerconsequence of (b) and (c): Mrówka's Theorem. (Tyconoff's Theorem yet again) . Each product of compact spaces is compact.
Proof.
In Theorem 2.1, take R to be class of topological spaces. References
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