As we have seen in§2.3.2, generalisations of the real line naturally lead to a generalisation of metrisability. The theory that we introduced is far from being complete.
In Theorem 2.40, we have seen that every λ-additive weakly λ-Choquet space is λ-Baire. We do not know if this is actually a characterisation.
Question 2.61. Is everyλ-additive λ-Baire space weaklyλ-Choquet?
By Theorem 2.35 every λ-Polish subspace of a λ-metrisable space isλ-Gδ. We still do not know if this implication can be reversed.
Question 2.62. Is everyλ-Gδ subset of aλ-metrisable set a λ-Polish space?
Moreover, as we mentioned in §2.3.1, we do not even know if every open subset of a λ-Polish space is λ-Polish.
Question 2.63. Is every open subset of aλ-metrisable set aλ-Polish space?
We believe that the theory of uniform spaces used in [2, 89, 97] is going to be central in answering the previous question.
Question 2.64. Is the sum of<λ many λ-Polish spaces λ-Polish?
Question 2.65. Is the product of <λ many λ-Polish spaces λ-Polish?
In [22] Coskey and Schlicht show that as in the classical case strongly λ-Choquet spaces are continuous images of generalised Baire spaceλλ. We do not know if this result can be proved for λ-Polish spaces.
Question 2.66. Is everyλ-Polish space a continuous image ofλλ?
In §2.3.1 we pointed out that we do not know the current status of the diagram on p. 25. In particular:
Question 2.67. Does Theorem 2.11 generalise to strongly λ-Polish spaces?
As we already remarked we do not know the large cardinal strength of Theorem 2.51.
Question 2.68. Can the assumption of λ being weakly compact be removed from The-orem 2.51?
In the classical theory, Theorem 2.51 is actually a characterisation of strongly Choquet spaces. We do not know if this is also true in the generalised case.
Question 2.69. Let Y be a λ-Gδ subspace of a strongly Choquet space. Is Y strongly Choquet?
Chapter 3
The generalised reals:
Bolzano-Weierstraß and Heine-Borel
Remarks on co-authorship. The results of this chapter are, unless otherwise stated, due to a collaboration of the author with Merlin Carl and Benedikt L¨owe. The results in §§3.2 & 3.3 appear in [16]. The questions and results in §3.4 are due solely to the author.
3.1 Introduction
Some properties do not transfer between R and ωω. One such property is the Bolzano-Weierstraß theorem BWT, i.e., “every sequence with bounded range has a cluster point”.
The propertyBWTconcerns the interplay between boundedness and sequential compact-ness, i.e., the relation between the order and the topology. Hence, the validity of BWT is not a purely topological property: it is not preserved by homeomorphisms and, more-over,BWTfails onωω.1 Another fundamental property of the real line is theHeine-Borel theorem HBT, i.e., “for every subsetX of R we have that X is compact if and only ifX is closed and bounded”. The BWT and the HBT are closely related: for ordered fields K, K is Dedekind-complete if and only if K satisfiesBWT if and only ifK satisfiesHBT (see, e.g., [73, Chapter 5, Theorem 7.6]). As well as theBWT, theHBTis also a property which is not preserved by homeomorphism. In particular, it does not transfer from Rto ωω.2
As mentioned, BWT and HBT both fail on Baire space, so the natural setting for uncountable generalisations of these theorems would not beκκ, but rather a generalisation of the real line. We will therefore focus on the status of these properties on the real ordinal numbers and on the generalised real line.
In the classical setting, the Bolzano-Weierstraß theorem is closely related to K˝onig’s lemma. This relationship was made precise by Harvey Friedman in the context of reverse mathematics. In reverse mathematics, theories in the language of second order arithmetic are used to compare the strength of classical theorems from everyday mathematics. In
1Letx(n)be the sequence (0n0. . .). The sequence (x(n); n∈ω) is bounded inωω, but has no cluster point.
2The clopen set [01] is bounded but not compact.
[32], Friedman studies extensions of the recursive comprehension axiom system (RCA); see [32, §I] for the definition ofRCA. In particular, in [32, Theorem 1.1], Friedman considers systems in which RCA is augmented with KL and BWT, respectively, and proves that these two systems are equivalent.
In the setting of Weihrauch reducibility the relationship between BWT andWKL was studied by Brattka, Gherardi, and Marcone [12]; they introduce a purely topological version of Bolzano-Weierstraß, BWTtop, i.e., “every sequence whose range has compact closure has a cluster point”. If a space X satisfies the BWTthen it satisfies the BWTtop. Lemma 3.1. Let X be a totally ordered set and (X, τ) be the order topology on X. If the property BWT holds in X, then the property BWTtop holds in (X, τ).
Proof. Let (xα)α∈λ as in the statement of the BWTtop. It is enough to prove that the sequence is bounded. Consider the following set of intervals C = {(x, y) | x, y ∈ X ∧ x < y}. The set C is a covering of the closure of the range of (xα)α∈λ. But then there are finitely many intervals (x0, y0), . . . ,(xn, yn) covering the closure of the range of the sequence. But then the range of the sequence is contained in the open interval (inf0≤i≤nxi,sup0≤i≤nyi). Therefore, the sequence is bounded.
In contrast to BWT, the property BWTtop holds in Baire space (the failure of BWT inωω corresponds to the fact that not all bounded subsets ofωω have compact closure).
Lemma 3.2. The BWTtop property holds in Baire space.
Proof. Let (xα)α∈λ be a sequence as in the statement of BWTtop. We want to find a cluster point s of the range of the sequence. We will define s by recursion. Note that the set C0 :={[n] ;, n ∈ω} is an open cover of the closure of the range of the sequence.
Therefore there must be a finite subcover of C0 and a natural number n0 ∈ω such that [n0] contains infinitely many points of the range of the sequence. Let s(0) := n0. Now, letC1 := (C0\ {[n0]})∪ {[n0n] ;n ∈ω}. The set C1 is again an open cover of the closure of the range of the sequence; and, as before, there are a finite subcover of C1 and a natural number n1 ∈ ω such that [n0n1] contains infinitely many points of the range of the sequence. Let s(1) := n1. In general assume that we have defined the sequence s up to m. Let Cm+1 := (Cm\ {[sm]})∪ {[(sm)n] ;n ∈ ω}. As before Cm+1 is again an open cover of the closure of the range sequence; therefore, there is a finite subcover of Cm+1 and a natural numbernm+1 ∈ω such that [(sm)nm+1] contains infinitely many points of the sequence. It is not hard to see that the sequence s ∈ ωω is a cluster point of (xα)α∈λ.
Writing BWTtopX for the statement “every sequence in X whose range has a compact closure has a cluster point in X”, Brattka, Gherardi, and Marcone proved:
BWTtop
R ≡W BWTtopωω ≡W WKL0,
where WKL0 denotes the jump of WKL. In the Weihrauch setting, the jump corresponds to an application of the monotone convergence theorem which allows us to do a transition from the subsequence produced byWKLto the cluster point needed byBWT;WKLis not sufficient to do that transition (in other words, WKL<W BWT). Note thatBWTtopX and
BWTX are not in general Weihrauch equivalent for arbitrary ordered spacesX; however, they are in the case X =R (because of HBT). Therefore, BWTR≡W WKL0.
In this chapter, we will discuss generalisations of BWT to uncountable cardinals κ.
For one of these, called the κ-weak Bolzano-Weierstraß theorem, we prove that if κ is inaccessible, then theκ-weak Bolzano-Weierstraß theorem holds for the generalised reals if and only if κ has the tree property (see Corollary 3.23) and the discussion on p. 34.
The chapter is organised as follows: in §3.2, we will study generalisations of the Bolzano-Weierstraß theorem on κ-R and Rκ; in §3.2.1 we will remind the reader of the classical Bolzano-Weierstraß theorem; in §3.2.2 we will study a generalised version of the Bolzano-Weierstraß theorem introduced by Sikorski; in§§3.2.3 and 3.2.4 we will introduce two natural version of generalised Bolzano-Weierstraß theorem and study their status on Rκ; finally, in §3.3, we will study of a generalised version of the Heine-Borel theorem.