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J-2 Consistency Results in Topology, I: Quotable Principles 419 J-2 Consistency Results in Topology, I: Quotable Principles hart v.2003/06/18 Prn:23/06/2003; 15:17 F:hartj02.tex; VTEX/EA p. 1 j-2 Consistency results in topology, I: Quotable principles 419 j-2 Consistency Results in Topology, I: Quotable Principles A noteworthy feature of general topology – as contrasted 1. The Continuum Hypothesis with geometry, number theory, and most other fields of mathematics – is that many of its fundamental questions are The best-known quotable principle is undoubtedly the Con- not decided by “the usual axioms of set theory”. What this tinuum Hypothesis, abbreviated as CH, which states that means is that if one codifies the set-theoretic assumptions the real line has minimum possible cardinality, i.e., c = ℵ used implicitly in ordinary mathematics and consolidates |R|=2 0 =ℵ1. It was proved consistent by Gödel in [5] these in a list of axioms – the usual axioms of set theory (see [Ku] for a proof) but before that numerous conse- – one is left with many statements that can neither be proved quences were derived from it, see, e.g., [7]. nor refuted on the basis of these axioms alone. In some It derives its strength from the facts that so many sets have sense such undecidability results are commonplace: every- cardinality c and the initial segments of ω1 are all countable; one knows that, say, the statement ϕ = (∀x)(∀y)(xy = yx) this makes for relatively easy transfinite constructions. To give the flavour we construct a Lusin set: an uncountable cannot be decided by “the usual axioms of Group Theory”. subset of R that meets every nowhere dense set in a count- Indeed, the group of integers is a model for “Group The- able set. It suffices to take an enumeration N : α<ω of ory plus ϕ”, which makes this conjunction consistent and α 1 the family of all closed nowhere dense sets and apply the ϕ non-refutable; the permutation group S3 does the same Baire Category Theorem each time to choose lα outside for ¬ϕ. { : } β<α Nα;then lα α<ω1 is the Lusin set. A similar con- A convenient list of axioms for Set Theory, one which struction will yield a strongly infinite-dimensional subspace has become standard is “ZFC” – Zermelo–Fraenkel set the- of the Hilbert cube all of whose finite-dimensional subsets ory, including the Axiom of Choice. See, e.g., [Ku] for a are countable. In [KV, Chapter 8, § 4] one finds more intri- detailed exposition. Thus there are important topological cate constructions, involving CH,ofhereditarily separable statements ϕ such that neither ϕ nor its negation follow nonLindelöf spaces (S-spaces)aswellashereditarily Lin- from ZFC. Establishing this is not as easy as in the case of delöf nonseparable spaces (L-spaces). Group Theory. While it is easy to prove the consistency of On occasion CH helps simply by counting: in combination ℵ ℵ Group Theory by providing models (groups) for it, the same with Jones’ Lemma CH (or even its consequence 2 0 < 2 1 ) is, by Gödel’s Second Incompleteness Theorem, in principle implies that in a separable normal space closed and dis- impossible for ZFC (or any other useful collection of axioms crete subsets are countable and hence that separable normal for set theory). Gödel’s theorem says that a consistency proof Moore spaces are metrizable. By contrast, CH implies there of a theory as strong as ZFC cannot be formalized in the the- is a non-metrizable normal Moore space [KV, Chapter 16]. ory itself. This explains why many results are formulated as Another counting example comes from the theory of uni- “if ZFC is consistent then so is ZFC plus ϕ” instead of sim- form spaces. It is well-known that in a uniform space the ply “ϕ is consistent with ZFC” and why, formally, we should finite uniform covers generate a uniformity again as do the be speaking of relative consistency proofs. Even if one is not countable uniform covers. The proofs can be generalized to ℵ interested in consistency results per se, it is nonetheless pru- show that any 1-sized uniform cover has a uniform star re- finement of cardinality c. Thus, CH implies that the ℵ -sized dent to be aware of them, lest one waste effort trying to prove 1 uniform covers generate a uniformity. In general, the Gen- a proposition that has a consistent negation. eralized Continuum Hypothesis (GCH), which states that There are two ways in which consistency results in to- + for all cardinals κ one has 2κ = κ , implies that for every κ pology are obtained. The first consists of proving implica- the uniform covers of cardinality less than κ generate a uni- tions between statements, where the antecedent is a state- formity. ment previously proven consistent with ZFC. Thus, the con- The Continuum Hypothesis also implies that there is an sequent itself is proven consistent with ZFC as well. The almost disjoint family of uncountable subsets of ω1 of car- ℵ present article deals with results like these and in partic- dinality 2 1 or, equivalently, that the cellularity of the space ℵ ular with the best-known combinatorial principles that oc- U(ω1) of uniform ultrafilters is the maximum possible 2 1 . cur as antecedents. In the second kind of consistency result Here ‘almost disjoint’ means that intersections are countable will be dealt with in the companion article; here one needs and an ultrafilter u on a set X is uniform if every element an intimate knowledge of how one actually proves (relative) has full cardinality, i.e., |U|=|X| for all U ∈ u. consistency results. The principal subjects in that article are See also the article on βN and βR for other applications forcing and large cardinals. of CH. hart v.2003/06/18 Prn:23/06/2003; 15:17 F:hartj02.tex; VTEX/EA p. 2 420 Section J: Influences of other fields 2. Martin’s Axiom less than c are meager and of measure zero. We discuss the consequences of CH mentioned above. Every consequence of CH is a potential theorem, as it can no Lusin sets no longer exist as every set of reals of cardi- longer be refuted. Some consequences are in fact equivalent nality less than c is of first category. It becomes harder to to CH, e.g., the existence of the special infinite-dimensional find S-andL-spaces. Indeed, MA +¬CH denies the exis- space above; such consequences are then automatically non- tence of compact such spaces and it is even consistent with provable because Cohen proved the consistency of the nega- MA +¬CH that no S-spaces exist. tion of CH (see the next article for an indication of the proof On the positive side: MA +¬CH implies that separable and [Ku] for a full treatment). To decide other consequences normal nonmetrizable Moore spaces exist. It also implies one would need proofs that avoid CH or principles that imply that there is a uniform space with a uniform cover of car- their negations. dinality ℵ1 without an ℵ1-sized uniform star refinement, to Martin’s Axiom (MA) is such a principle. It can be wit the subspace {x:x=1 (∀α)(xα 0)} of ∞(ℵ1) [6]. viewed as an extension of the Baire Category Theorem: it The latter paper does not mention MA +¬CH directly but the states that if X is a compact Hausdorff space with the count- proof needs a family A,ofsizeℵ2, of uncountable subsets able chain condition (the ccc for short) then the union of of ω1 such that for some fixed cardinal κ ℵ2 every sub- fewer than c many nowhere dense in X still has a dense com- family A of A of size κ has a finite intersection. To make plement. As such it is a consequence of CH but the conjunc- such a family one starts with an almost disjoint family B,of tion MA +¬CH is also consistent. size ℵ2, of uncountable subsets of ω1 (almost disjoint means Its original formulation, though more extensive, is ulti- distinct elements have a countable intersection); by repeated mately more useful: if P is a partially ordered set (a poset) application of [4, 42I] one shrinks the elements of B to pro- with the ccc and D is a family of fewer than c dense sets then duce the desired family A (with κ = 2). there is a filter G on P that meets all members of D.Twoel- The ccc is the weakest in a line of properties, the best- ements p and q of P are compatible if there is an element r known of these are σ -centered (corresponding to separa- with r p,q,andincompatible otherwise. An antichain is ble compact spaces) and countable (corresponding to com- a set of mutually incompatible elements and “P has the ccc” pact metrizable spaces). These in turn give rise to weaken- means every antichain is countable. A set D ⊆ P is a dense ings of MA that have generated interest of their own, since set if for every p there is d ∈ D with d p. Finally, a filter they provide the possibility of denying some of MAs conse- on P is a subset G that satisfies: if p,q ∈ G then there is quences while retaining others. r ∈ G with r p,q,andifp ∈ G and q p then q ∈ G. Martin’s Axiom for σ -centered partially ordered sets The equivalence between the two formulations becomes was shown to be equivalent to the purely combinatorial state- somewhat apparent when one thinks of P as representing the ment known variously as P(c) or p = c:ifF is a family, of open sets of the space X – a proof may be devised along cardinality less than c, of subsets of N with the strong finite the lines of the Stone Representation Theorem of Boolean intersection property, i.e., the intersection of every finite algebras.
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