Generic left-separated spaces and calibers

arXiv (Cornell University), 2023

We show that the definition of caliber given by Engelking in [5], which we will call caliber*, differs from the traditional notion of this concept in some cases and agrees in others. For instance, we show that if κ is an infinite cardinal with 2 κ < ℵκ and cf(κ) > ω, then there exists a compact Hausdorff space X such that o(X) = 2 ℵκ = |X|, ℵκ is a caliber* for X and ℵκ is not a caliber for X. On the other hand, we obtain that if λ is an infinite cardinal number, X is a Hausdorff space with |X| > 1, φ ∈ {w, nw}, o(X) = 2 φ(X) and µ := o X λ , then the calibers of X λ and the true calibers* (that is, those which are less than or equal to µ) coincide, and are precisely those that have uncountable cofinality.

Topology and its Applications, 2010

We obtain from the consistency of the existence of a measurable cardinal the consistency of "small" upper bounds on the cardinality of a large class of Lindelöf spaces whose singletons are G δ sets.

The Journal of Symbolic Logic, 2020

We study the consistency and consistency strength of various configurations concerning the cardinal characteristics $\mathfrak {s}_\theta , \mathfrak {p}_\theta , \mathfrak {t}_\theta , \mathfrak {g}_\theta , \mathfrak {r}_\theta $ at uncountable regular cardinals $\theta $ . Motivated by a theorem of Raghavan–Shelah who proved that $\mathfrak {s}_\theta \leq \mathfrak {b}_\theta $ , we explore in the first part of the paper the consistency of inequalities comparing $\mathfrak {s}_\theta $ with $\mathfrak {p}_\theta $ and $\mathfrak {g}_\theta $ . In the second part of the paper we study variations of the extender-based Radin forcing to establish several consistency results concerning $\mathfrak {r}_\theta ,\mathfrak {s}_\theta $ from hyper-measurability assumptions, results which were previously known to be consistent only from supercompactness assumptions. In doing so, we answer questions from [1], [15] and [7], and improve the large cardinal strength assumptions for results from ...

Acta Mathematica Academiae Scientiarum Hungaricae, 1969

Canadian Journal of Mathematics, 1976

Introduction. Our method using CH is a blend of two earlier constructions and Ostaszewski ) of hereditarily separable {HS), regular, non-Lindelôf, first countable spaces. [4] produces a much better space than ours in § 1 ; it has all of our properties except that it is not realcompact (which is probably more interesting), and it is countably compact as well; however, the construction works only under O, which implies the continuum hypothesis (CH) but is not equivalent to it. The argument of [2], like ours, just needs CH, but it is much more complicated, and it is not immediate that the space produced is locally compact or perfectly normal (although, in fact, it is; see the remark at the end of § 1). In § 2, we use a more complicated version of the technique in § 1 to construct a first countable, cardinality coi, HS, Dowker space. A Dowker space is a normal, Hausdorff space which is not countably paracompact. There is a known "real" Dowker space but all of its cardinal functions are large . There is a known HS Dowker space but its construction depends on the existence of a Souslin line . It was an old conjecture that the existence of a small cardinality (or small cardinal function) Dowker space depended on the existence of a Souslin line, and this conjecture is disproved by our construction. Using our technique and O (which implies both CH and the existence of a Souslin line) we can construct a first countable, cardinality coi, HS, Dowker space which is also locally compact and c-countably compact; but we choose the weaker hypothesis over the stronger conclusion. In § 2 we use Lusin sets in our construction. A subset L of the line is Lusin if L is uncountable and every nowhere dense subset of L is countable. If we assume CH, then there are Lusin sets in the line. However if we assume Martin's axiom and the negation of CH, then there are no Lusin sets in the line. If we assume Martin's axiom and the negation of CH, then there is no non-Lindelôf, first countable, regular topology on a subset of the line which refines the usual topology and has the property that the closure of a set in the two topologies differs by an at most countable set. Since our construction in § 1 yields just such a topology, both constructions are independent of the usual axioms for set theory. 1. The basic idea for obtaining this space is to start with the usual topology of the real numbers (R), which has many of the properties we want; in particular

Studia Scientiarum Mathematicarum Hungarica, 2007

A cardinal λ is called ω -inaccessible if for all µ &lt; λ we have µ ω &lt; λ . We show that for every ω -inaccessible cardinal λ there is a CCC (hence cardinality and cofinality preserving) forcing that adds a hereditarily Lindelöf regular space of density λ . This extends an analogous earlier result of ours that only worked for regular λ .

Topology and its Applications

A topological space X is called almost discretely Lindelöf if every discrete set D ⊂ X is included in a Lindelöf subspace of X. We say that the space X is µ-sequential if for every nonclosed set A ⊂ X there is a sequence of length ≤ µ in A that converges to a point which is not in A. With the help of a technical theorem that involves elementary submodels, we establish the following two results concerning such spaces. (1) For every almost discretely Lindelöf T 3 space X we have |X| ≤ 2 χ(X). (2) If X is a µ-sequential T 2 space of pseudocharacter ψ(X) ≤ 2 µ and for every free set D ⊂ X we have L(D) ≤ µ, then |X| ≤ 2 µ. The case χ(X) = ω of (1) provides a solution to Problem 4.5 of [5], while the case µ = ω of (2) is a partial improvement on the main result of [2]. Our main aim in this note is to prove what is stated in the title and thus give a solution to problem 4.5 of [5]. All spaces in here are assumed to be T 1. Consequently, if X is any space and A is any subset of X then the pseudocharacter ψ(A, X) of A in X, i.e. the smallest size of a family of open sets whose intersection is A, is well-defined. We recall that a transfinite sequence {x α : α < η} ⊂ X is called a free sequence in X if for every β < η we have {x α : α < β} ∩ {x α : β ≤ α < η} = ∅. We say that a subset D ⊂ X is free if it has a well-ordering that turns it into a free sequence. Clearly, every free set is discrete and

Topology and its Applications, 2017

It is well known that separation axioms together with some local and global cardinal invariants lead to restriction of the cardinality of a given topological space. For an extensive survey, one can look at [Hod06]. We shall mention here just a few such results that are directly related to this paper. At the end we will discuss some long-standing open problems in cardinal invariant theory and give a possible way of approaching a solution.

We investigate properties of the class of compact spaces on which every regular Borel measure is separable. This class will be referred to as MS. We discuss some closure properties ofMS, and show that some simply defined compact spaces, such as compact ordered spaces or compact scattered spaces, are in MS. Most of the basic theory for regular measures is true just in ZFC. On the other hand, the existence of a compact ordered scattered space which carries a non-separable (non-regular) Borel measure is equivalent to the existence of a real-valued measurable cardinal c. We show that not being in MS is preserved by all forcing extensions which do not collapse !1, while being in MS can be destroyed even by a ccc forcing.

Annals of Pure and Applied Logic

We introduce and analyze a new cardinal characteristic of the continuum, the splitting number of the reals, denoted s(R). This number is connected to Efimov's problem, which asks whether every infinite compact Hausdorff space must contain either a non-trivial convergent sequence, or else a copy of βN. Lemma 2.3. Suppose X is an uncountable, zero-dimensional, Borel subspace of a Polish space. Then s(X) = s(2 ω). Proof. If X is as in the statement of the lemma, then X contains a copy of the Cantor space [23, Theorem 6.2] and X embeds topologically into the Cantor space [23, Theorem 7.3]. By the previous lemma, this implies that s(2 ω) ≤ s(X) ≤ s(2 ω), so s(X) = s(2 ω). Lemma 2.4. s(2 ω) is uncountable. Proof. It follows immediately from the definitions that if N has the discrete topology, then s(N) is equal to the splitting number s (which is uncountable). As s ≤ s(2 ω) by Lemma 2.2, s(R) is uncountable. Theorem 2.5. If X, Y are uncountable Polish spaces, then s(X) = s(Y). Proof. Let H denote the Hilbert cube [0, 1] ℵ 0. To prove the theorem, it suffices to show that s(H) ≤ s(2 ω). This is because, if X is any uncountable Polish space, then X contains a copy of the Cantor space [23, Theorem 6.2] and X embeds into the Hilbert cube [23, Theorem 4.14]. By Lemma 2.2, it follows that s(2 ω) ≤ s(X) ≤ s(H), so if s(H) ≤ s(2 ω) then s(H) = s(2 ω) = s(X) for every uncountable Polish space X. To see that s(H) ≤ s(2 ω), we use a slight variation of a result of Hausdorff [20], which states that the Baire space ω ω can be written as an increasing union α<ω 1 X α , where each X α is a G δ subspace of ω ω. This result is a relatively straightforward consequence of the existence of Hausdorff gaps. Recall that a Hausdorff gap is a sequence (f α , g α) : α < ω 1 of pairs of functions ω → ω such that • f α < * f β < * g β < * g α for every α < β < ω 1 (where, as usual, f < * g means that f (n) < g(n) for all but finitely many n ∈ ω). • there is no h : ω → ω such that f α < * h < * g α for all α < ω 1. Taking X α = {f ∈ ω ω : f α < * f < * g α } for each α < ω 1 , one may check that each X α is G δ and that ω ω = α<ω 1 X α. Recall that ω ω is homeomorphic to [0, 1] \ Q, so we may write [0, 1] \ Q =