We characterize those regular continuous frames for which the least compactification is a perfect compactification. Perfect compactifications are those compactifications of frames for which the right adjoint of the compactification map preserves disjoint binary joins. Essential to our characterization is the construction of the frame analog of the twopoint compactification of a locally compact ...

We force and construct a model in which level by level equivalence between strong compactness and supercompactness holds, along with certain additional “inner model like” properties. In particular, in this model, the class of Mahlo cardinals reflecting stationary sets is the same as the class of weakly compact cardinals, and every regular Jonsson cardinal is weakly compact. On the other hand, w...

We show that if Part(κ, λ) holds for every λ ≥ κ, then κ is strongly compact. Let κ be a regular infinite cardinal, and let λ ≥ κ be a cardinal. Pκ(λ) denotes the set of all subsets of λ of size less than κ. Part(κ, λ) means that for every F : Pκ(λ) × Pκ(λ) → 2, there is a cofinal subset A of (Pκ(λ),⊆) such that F is constant on the set {(a, b) ∈ A × A : a ⊂ b}. This definition is due to Jech [...

We prove, in ZFC alone, some new results on regularity and decomposability of ultrafilters; among them: (a) If m ≥ 1 and the ultrafilter D is (im(λ),im(λ))regular then D is κ-decomposable for some κ with λ ≤ κ ≤ 2 (Theorem 4.3(a)). (b) If λ is a strong limit cardinal and D is (im(λ ),im(λ ))regular then either D is (cf λ, cf λ)-regular or there are arbitrarily large κ < λ for which D is κ-decom...

We present a characterization of weakly compact cardinals in terms of generalized stationarity. We apply this characterization to construct a model with no partial square sequences. One of the striking features of weak compactness in the large cardinal hierarchy is the wide variety of characterizations of the concept. These characterizations meet many disparate areas of set theory, including me...

Let H 0(X) (H(X)) denote the set of all (nonempty) closed subsets of X endowed with the Vietoris topology. A basic problem concerning H(X) is to characterize those X for which H(X) is countably compact. We conjecture that u-compactness of X for some u ∈ ω∗ (or equivalently: all powers of X are countably compact) may be such a characterization. We give some results that point into this direction...

in this paper‎, ‎we introduce a new model of point free topology‎ ‎for point-set topology‎. ‎we study the basic concepts in this‎ ‎structure and will find that it is naturally close to point-set‎ ‎topology‎.‎

The aim of this paper is to consider questions concerning the possible maximum cardinality various separable pseudoradial (in short: SP) spaces. most intriguing question here if there in ZFC a regular (or just Hausdorff) SP space >c. While left open, we establish number non-trivial results that list below. It consistent with MA+c=ℵ2 countably tight and compact 2c. If κ measurable cardinal then ...

Let κ be an uncountable regular cardinal. Call an equivalence relation on functions from κ into 2 Σ11-definable over H(κ) if there is a first order sentence φ and a parameter R ⊆ H(κ) such that functions f, g ∈ κ2 are equivalent iff for some h ∈ κ2, the structure 〈H(κ),∈, R, f, g, h〉 satisfies φ, where ∈, R, f , g, and h are interpretations of the symbols appearing in φ. All the values μ, 1 ≤ μ...

Classically, a Tychonoff space is called strongly 0-dimensional if its Stone-Čech compactification is 0-dimensional, and given the familiar relationship between spaces and frames it is then natural to call a completely regular frame strongly 0-dimensional if its compact completely regular coreflection is 0-dimensional (meaning: is generated by its complemented elements). Indeed, it is then seen...