summary:Many fundamental mathematical results fail in {\bf{ZF}}, i.e., in Zermelo-Fraenkel set theory without the Axiom of Choice. This article surveys results --- old and new --- that specify how much ``choice'' is needed {\it precisely} to validate each of certain basic analytical and topological results

Herrlich, Horst

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Institute of Mathematics AS CR, v. v. i.

Commentationes Mathematicae Universitatis CarolinaeHorst HerrlichChoice principles in elementary topology and analysisCommentationes Mathematicae Universitatis Carolinae, Vol. 38 (1997), No. 3, 545--552Persistent URL: http://dml.cz/dmlcz/118951Terms of use:© Charles University in Prague, Faculty of Mathematics and Physics, 1997Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document mustcontain these Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://project.dml.czComment.Math.Univ.Carolin. 38,3 (1997)545–552 545Choice principles in elementary topology and analysisHorst HerrlichKarl Peter Grotemeyer, meinem verehrten Lehrer, zum 70. Geburtstag gewidmetAbstract. Many fundamental mathematical results fail in ZF, i.e., in Zermelo-Fraenkelset theory without the Axiom of Choice. This article surveys results — old and new— that specify how much “choice” is needed precisely to validate each of certain basicanalytical and topological results.Keywords: Axiom of (Countable) Choice, Boolean Prime Ideal Theorem, Theorems ofAscoli, Baire, Čech-Stone and Tychonoff, compact, Lindelöf and orderable spacesClassification: 03E25, 04A25, 26A03, 26A15, 54-02, 54A35, 54B10, 54C35, 54D20,54D30, 54D65, 54E45, 54E50, 54E52Introduction“It is a peculiar fact that all the transfinite axiomsare deducible from a single one, the axiom of choice,— the most challenged axiom in the mathematicalliterature.”David Hilbert (1926)“A formal system in which ∃G(x) is provable, butwhich provides no method for finding the x in ques-tion, is one in which the existential quantifier failsto fulfill its intended function.”A.H. Goldstein (1968)“The Axiom of Choice has easily the most torturedhistory of all the set-theoretic axioms.”Penelope Maddy (1988)As the above quotes indicate the Axiom of Choice is highly controversial —perhaps the most controversial axiom in all of mathematics. To discuss this issuehowever is not the purpose of the present note, — see instead the inspiring bookby G.H. Moore (1982) –, its more modest aim is to demonstrate how much“choice” is needed to make elementary analysis resp. topology click.The results (all in ZF) are ordered by their degree of abstractness.546 H.Herrlich1. In the realm of the realsWe start by observing that several familiar topological properties of the reals areequivalent to each other and to rather natural choice-principles.Theorem 1.1 ([15], [29], [30]). Equivalent are:1. in R, a point x is an accumulation point of a subset A iff there exists asequence in A \ {x} that converges to x,2. a function f : R → R is continuous at a point x iff it is sequentially con-tinuous at x,3. a real-valued function f : A → R from a subspace A of R is continuous iffit is sequentially continuous,4. each subspace of R is separable,5. R is a Lindelöf space,6. Q is a Lindelöf space,7. N is a Lindelöf space,8. each unbounded subset of R contains an unbounded sequence,9. the Axiom of Choice for countable collections of subsets of R.There exist models of ZF that violate the above conditions ([17], [18]). Ob-serve the fine distinction between conditions 2 and 3 of Theorem 1.1. These maylead one to assume that also the following property is equivalent to the aboveconditions:(*) a function f : R −→ R is continuous iff it is sequentially continuous.However, this would be a serious mistake: (*) holds in ZF (without any choice-assumptions) — see [29]. If, however, we consider functions f : X −→ R withmetric domain we need even more choice than in Theorem 1.1, — see Theorem 2.1.Proposition 1.2 ([15]). Equivalent are:1. in R, every bounded infinite set contains a convergent injective sequence,2. every infinite subset of R is Dedekind-infinite.There exist models of ZF that violate the above conditions ([18]).Obviously, the conditions of Theorem 1.1 imply the conditions of Proposi-tion 1.2.Is the converse true?Observe that the following slight modifications of condition 1 in Proposition 1.2hold in ZF:(a) in R, every bounded countable set contains a convergent injective se-quence,(b) in R, for every bounded infinite set there exists an accumulation point.2. In the realm of pseudometric spacesIn this section we consider (pseudo)metric spaces and various compactness-notionsfor them.Choice principles in elementary topology and analysis 547Theorem 2.1 ([4], [15]). Equivalent are:1. every separable pseudometric space is a Lindelöf space,2. every pseudometric space with a countable base is a Lindelöf space,3. the Axiom of Choice for countable collections of subsets of R.Definition 2.2. A pseudometric space X is called1. Heine-Borel-compact provided every open cover ofX contains a finite one,2. Weierstraß-compact provided for every infinite subset of X there existsan accumulation point,3. Alexandroff-Urysohn-compact provided for every infinite subset of Xthere exists a complete accumulation point,4. sequentially-compact provided every sequence in X has a convergent sub-sequence.Under the Axiom of Choice the above compactness concepts are equivalent.This is no longer the case in ZF.However the following implications remain valid ([4]):AAlexandroff-Urysohn-compact Heine-Borel-compactWeierstraß-compact complete and totally boundedsequentially compactTheorem 2.3 ([4]). Equivalent are:1. Weierstraß-compact⇔ sequentially compact,2. finite ⇔ Dedekind-finite.Theorem 2.4 ([4], [14]). Equivalent are:1. in a (pseudo)metric space X , a point x is an accumulation point of asubset A iff there exists a sequence in A \ {x} that converges to x,2. a real-valued function f : X→ R from a (pseudo)metric space is continu-ous iff it is sequentially continuous,3. a function between (pseudo)metric spaces is continuous iff it is sequentiallycontinuous,4. subspaces of a separable pseudometric spaces are separable,548 H.Herrlich5. totally bounded pseudometric spaces are separable,6. Heine-Borel-compact pseudometric spaces are separable,7. separable ⇔ countable base,8. separable ⇔ Lindelöf,9. separable ⇔ topologically totally bounded,10. Heine-Borel-compact ⇔ sequentially compact,11. Heine-Borel-compact ⇔ complete and totally bounded,12. sequentially compact ⇔ complete and totally bounded,13. Weierstraß-compact⇔ complete and totally bounded,14. complete ⇔ each Cauchy-filter converges,15. the Baire Category Theorem for complete, totally bounded pseudometricspaces,16. the Baire Category Theorem for complete pseudometric spaces withcountable base,17. the Axiom of Countable Choice.Observe that the Baire Category Theorem for complete, separable pseudomet-ric spaces holds in ZF. However, the role of the Baire Category theorem for allcomplete pseudometric spaces is not yet clear:The Axiom of Dependent Choices implies the Baire Category Theorem for com-plete pseudometric spaces, and the latter implies the Axiom of Countable Choice.Is one of these implications an equivalence?Theorem 2.5 ([4], [10]). Equivalent are:1. Alexandroff-Urysohn-compact⇔ Heine-Borel-compact,2. Alexandroff-Urysohn-compact⇔ Weierstraß-compact,3. Alexandroff-Urysohn-compact⇔ sequentially compact,4. the Axiom of Choice.3. In the realm of topological spacesFinally, we consider arbitrary topological spaces and various familiar compactnessnotions there — all equivalent under the Axiom of Choice.Definition 3.1. A topological space X is called1. Heine-Borel-compact provided every open cover ofX contains a finite one,2. Alexandroff-Urysohn-compact provided for every infinite subset of Xthere exists a complete accumulation-point,3. Bourbaki-compact provided every ultrafilter converges in X,4. Comfort-compact provided X is homeomorphic to a closed subspace of[0, 1]I for some I.Theorem 3.2([15]). Equivalent are:1. every topological space with a countable base is a Lindelöf space,2. the Axiom of Choice for countable collections of subset of the reals.Choice principles in elementary topology and analysis 549Theorem 3.3 ([1], [3], [10], [13], [14], [21], [32]). Equivalent are:1. the Tychonoff Theorem for Heine-Borel-compact spaces,2. the Tychonoff Theorem for Alexandroff-Urysohn-compact spaces,3. the Čech-Stone Theorem1 for Alexandroff-Urysohn-compactness,4. the Ascoli-Theorem for Alexandroff-Urysohn-compactness,5. products of spaces with finite topologies are Heine-Borel-compact,6. products of spaces with finite topologies are Alexandroff-Urysohn-compact,7. products of finite discrete spaces are Alexandroff-Urysohn-compact,8. spaces with finite topologies are Alexandroff-Urysohn-compact,9. finite products of Alexandroff-Urysohn-compact spaces are Alexandroff-Urysohn-compact,10. finite coproducts of Alexandroff-Urysohn-compact spaces areAlexandroff-Urysohn-compact,11. Heine-Borel-compact ⇔ Alexandroff-Urysohn-compact,12. Bourbaki-compact⇔ Alexandroff-Urysohn-compact,13. Comfort-compact ⇔ Alexandroff-Urysohn-compact and completelyregular,14. the Axiom of Countable Choice and every closed filter is contained ina maximal one,15. the Axiom of Choice.The Axiom of Dependent Choices implies the Tychonoff Theorem for count-able collections of Heine-Borel-compact spaces, and the latter implies the Axiomof Countable Choice ([7]). Is one of these implications an equivalence? It isknown however that the Axiom of Countable Choice does not imply the Axiomof Dependent Choices ([20]).Theorem 3.4 ([2], [3], [10], [12], [14], [22], [25], [28]). Equivalent are:1. the Tychonoff Theorem for Heine-Borel-compact Hausdorff spaces,2. the Čech-Stone Theorem1 for Heine-Borel-compactness,3. the Ascoli-Theorem for Heine-Borel-compactness,4. the Ascoli-Theorem for Bourbaki-compactness,5. the Ascoli-Theorem for Comfort-compactness,6. products of finite spaces are Heine-Borel-compact,7. Heine-Borel-compact ⇔ Bourbaki-compact,8. Heine-Borel-compact and completely regular ⇔ Comfort-compact,9. every z-filter is fixed ⇔ every z-ultrafilter is fixed,10. in the ring C(X), every ideal is fixed iff every maximal ideal is fixed,1The Čech-Stone Theorem states that compact Hausdorff spaces form an epireflective sub-category of the category of Hausdorff spaces and continuous maps (i.e., that for every Hausdorffspace X there exists a compact Hausdorff space βX and a continuous dense map r : X −→ βXsuch that for every compact Hausdorff space Y and every continuous map f : X −→ Y thereexists a continuous map f̄ : βX −→ Y with f = f̄ ◦ r).550 H.Herrlich11. in the ring C∗(X), every ideal is fixed iff every maximal ideal is fixed,12. every filter is contained in an ultrafilter,13, every zero-filter is contained in a maximal one,14. the Boolean Prime Ideal Theorem.The Boolean Prime Ideal Theorem is known to be properly weaker than theAxiom of choice [9]. This is a deep result, but the gap — surprisingly — is evenwider. The Boolean Prime Ideal Theorem together with the Axiom of CountableChoice (or even with the Axiom of Dependent Choices) is properly weaker thanthe Axiom of Choice ([26]).Theorem 3.5([10]). Equivalent are:1. the Tychonoff Theorem for Bourbaki-compact spaces,2. either the Axiom of Choice or all ultrafilters are fixed.This result is particularly surprising and rather strange indeed, since the twoparts of condition 2 are rather far apart. The second part is not void, as onemight suspect, since there exist models of ZF with no free ultrafilters ([5]).Proposition 3.6 ([10]). Equivalent are:1. Alexandroff-compactifications of discrete spaces are Alexandroff-Urysohn-compact,2. finite ⇔ Dedekind-finite.The condition “finite ⇔ Dedekind-finite” of the above proposition is properlyweaker than the Axiom of Countable Choice ([19]). This is perhaps not surprising,but rather hard to prove.Theorem 3.7 ([7], [31]). Equivalent are:1. Urysohn’s Lemma for orderable2 spaces,2. Tietze-Urysohn’s Extension Theorem for orderable2 spaces,3. orderable2 spaces are normal,4. orderable2 spaces are collectionwise normal,5. orderable2 spaces are collectionwise Hausdorff,6. orderable2 spaces are monotonically normal,7. completely orderable2 spaces are hereditarily normal,8. the Axiom of Choice for collections of pairwise disjoint convex open subsetsof some completely ordered2 set,9. the Axiom of Choice for the collection of all non-empty convex subsets ofsome completely ordered2 set.Let us sum things up: Topology with “choice” may be as unreal as a soap-bubble dream, but topology without “choice” is as horrible as nightmare.2Here order means linear order.Choice principles in elementary topology and analysis 551References[1] Alas O.T., The Axiom of choice and two particular forms of Tychonoff theorem, Portugal.Math. 28 (1968), 75–76.[2] Banaschewski B., Compactification and the axiom of choice, unpublished manuscript, 1979.[3] Bentley H.L., Herrlich H., Compactness and rings of continuous functions — without theaxiom of choice, to appear.[4] Bentley H.L., Herrlich H., Countable choice and pseudometric spaces, to appear.[5] Blass A., A model without ultrafilters, Bull. Acad. Sci. Polon., Sér. Sci. Math. Astr. Phys.25 (1977), 329–331.[6] Comfort W.W., A theorem of Stone-Čech type, and a theorem of Tychonoff type, withoutthe axiom of choice; and their realcompact analogues, Fund. Math. 63 (1988), 97–110.[7] Good C., Tree I.J., Continuing horrors of topology without choice, Topol. Appl. 63 (1995),79–90.[8] Goodstein R.L., Existence in Mathematics, in: Logic and Foundations of Mathematics (eds.D. van Dalen et al.), Wolters-Noordhoff Publ. Co., 1968, pp. 70–82.[9] Halpern J.D., Lévy A., The Boolean prime ideal theorem does not imply the axiom ofchoice, Proc. of Symposium Pure Math. of the AMS 13 (1971), Part I, 83–134.[10] Herrlich H., Compactness and the Axiom of Choice, Appl. Categ. Structures 4 (1996),1–14.[11] Herrlich H., An effective construction of a free ultrafilter, Papers on Gen. Topol. Appl.(eds. S. Andima et al.), Annals New York Acad Sci. 806 (1996), 201–206.[12] Herrlich H., The Ascoli Theorem is equivalent to the Boolean Prime Ideal Theorem, toappear.[13] Herrlich H., The Ascoli Theorem is equivalent to the Axiom of Choice, to appear.[14] Herrlich H., Steprāns J., Maximal filters, continuity and choice principles, to appear inQuaestiones Math. 20 (1997).[15] Herrlich H., Strecker G.E., When in N Lindelöf?, to appear in Comment. Math. Univ.Carolinae 38 (1997).[16] Hilbert D., Über das Unendliche, Mathem. Annalen 95 (1926), 161–190.[17] Jaegermann M., The Axiom of Choice and two definitions of continuity, Bull. Acad. Polon.Sci, Sér. Sci, Math., Astr. et Phys. 13 (1965), 699–704.[18] Jech T., Eine Bemerkung zum Auswahlaxiom, Časopis Pěst. Mat. 93 (1968), 30–31.[19] Jech T.J., The Axiom of Choice, North Holland, Amsterdam, 1973.[20] Jensen R.B., Independence of the Axiom of Dependent Choices from the Countable Axiomof Choice, J. Symb. Logic 31 (1966), 294.[21] Kelley J.L., The Tychonoff product theorem implies the axiom of choice, Fund. Math. 37(1950), 75–76.[22] Loś J., Ryll-Nardzewski C., Effectiveness of the representation theory for Boolean algebras,Fund. Math. 41 (1955), 49–56.[23] Maddy P., Believing the axioms I., J. Symb. Logic 53 (1988), 481–511.[24] Moore G.H., Zermelo’s Axiom of Choice. Its Origins, Developments and Influence, Sprin-ger, New York, 1982.[25] Mycielski J., Two remarks on Tychonoff’s product theorem, Bull. Acad. Polon. Sci. Sér.Sci. Math., Astr. Phys. 12 (1964), 439–441.[26] Pincus D., Adding Dependent Choice to the Boolean Prime Ideal Theorem, Logic Colloq.76 (1977), 547–565.[27] Rubin H., Rubin J.E., Equivalents of the Axiom of Choice II., North Holland, Amsterdam,1985.[28] Rubin H., Scott D., Some topological theorems equivalent to the Boolean prime ideal the-orem, Bull. Amer. Math. Soc. 60 (1954), 389.[29] Sierpin̄ski W., Sur le rôle de l’axiome de M. Zermelo dans l’Analyse moderne, Compt.Rendus Hebdomadaires des Sēances de l’Academie des Sciences, Paris 193 (1916), 688–691.552 H.Herrlich[30] Sierpin̄ski W., L’axiome de M. Zermelo et son rôle dans la théorie des ensembles etl’analyse, Bull. Acad. Sci. Cracovie, Cl. Sci. Math., Sér. A (1918), 97–152.[31] Douwen E.K., Horrors of topology without AC: a nonnormal orderable space, Proc. Amer.Math. Soc. 95 (1985), 101–105.[32] Ward L.E., A weak Tychonoff theorem and the axiom of choice, Proc. Amer. Math. Soc.13 (1962), 757–758.University of Bremen, FB 3, P.O. Box 330440, 28334 Bremen, GermanyE-mail : herrlich@informatik.Uni-Bremen.de(Received April 3, 1997, revised June 5, 1997)

Choice principles in elementary topology and analysis

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Institute of Mathematics AS CR, v. v. i.