In this paper we use a natural forcing to construct a left-separated topology on an arbitrary cardinal κ. The resulting left-separated space Xκ is also 0-dimensional T2, hereditarily Lindelöf, and countably tight. Moreover if κ is regular then d(Xκ) = κ, hence κ is not a caliber of Xκ, while all other uncountable regular cardinals are. This implies that some results of [A] and [JSz] are, consis...

We say that a 〈∨, 0〉-semilattice S is conditionally co-Brouwerian, if (1) for all nonempty subsets X and Y of S such that X ≤ Y (i.e., x ≤ y for all 〈x, y〉 ∈ X × Y ), there exists z ∈ S such that X ≤ z ≤ Y , and (2) for every subset Z of S and all a, b ∈ S, if a ≤ b ∨ z for all z ∈ Z, then there exists c ∈ S such that a ≤ b ∨ c and c ≤ Z. By restricting this definition to subsets X , Y , and Z ...

We show that either of the following hypotheses imply that there is an inner model with a proper class of strong cardinals and a proper class of Woodin cardinals. 1) There is a countably closed cardinal κ ≥ א3 such that κ and (κ) fail. 2) There is a cardinal κ such that κ is weakly compact in the generic extension by Col(κ, κ). Of special interest is 1) with κ = א3 since it follows from PFA by ...

We prove the following: (1) If κ is weakly inaccessible then NSκ is not κ+-saturated. (2) If κ is weakly inaccessible and θ < κ is regular then NSθ κ is not κ +saturated. (3) If κ is singular then NS κ+ is not κ++-saturated. Combining this with previous results of Shelah, one obtains the following: (A) If κ > א1 then NSκ is not κ+-saturated. (B) If θ+ < κ then NSθ κ is not κ +-saturated.

We are proving the following: (1) If κ is a weakly inaccessible then NSκ is not κ -saturated. (2) If κ is a weakly inaccessible and θ < κ is regular then NS κ is not κ saturated. (3) If κ is singular then NS κ+ is not κ-saturated. Combining this with previous results of Shelah, one obtains the following: (A) If κ > א1 then NSκ is not κ -saturated. (B) If θ < κ then NS κ is not κ -saturated.

Suppose V ⊆ W are models of ZFC with the same ordinals, and that for all regular cardinals κ in W , V satisfies κ. If W \ V contains a sequence r : ω → γ for some ordinal γ, then for all cardinals κ < λ in W with κ regular in W and λ ≥ γ, (Pκ(λ)) \ V is stationary in (Pκ(λ)) . That is, a new ω-sequence achieves global co-stationarity of the ground model.

Let κ be a regular cardinal. Consider the Baire numbers of the spaces (2)κ for various θ ≥ κ. Let l be the number of such different Baire numbers. Models of set theory with l = 1 or l = 2 are known and it is also known that l is finite. We show here that if κ > ω, then l could be any given finite number. The Baire number of a topological space with no isolated points is the minimal cardinality ...

A universal space is one that continuously maps onto all others of its own kind and weight. We investigate when a universal Uniform Eberlein compact space exists for weight κ. If κ = 2<κ, then they exist whereas otherwise, in many cases including κ = ω1, it is consistent that they do not exist. This answers (for many κ and consistently for all κ) a question of Benyamini, Rudin and Wage of 1977.

We construct models containing exactly one supercompact cardinal in which level by level inequivalence between strong compactness and supercompactness holds. In each model, above the supercompact cardinal, there are finitely many strongly compact cardinals, and the strongly compact and measurable cardinals precisely coincide. Say that a model containing supercompact cardinals satisfies level by...