If κ < λ are such that κ is both supercompact and indestructible under κ-directed closed forcing which is also (κ+,∞)-distributive and λ is 2λ supercompact, then by [3, Theorem 5], {δ < κ | δ is δ+ strongly compact yet δ isn’t δ+ supercompact} must be unbounded in κ. We show that the large cardinal hypothesis on λ is necessary by constructing a model containing a supercompact cardinal κ in whic...

Questions regarding the continuity in κ of SLEκ traces and maps appear very naturally study SLE. In order to first question, we consider a natural coupling SLE traces: for different values use same Brownian motion. It is assume that with probability one, depends continuously on κ. rather easy show continuous Carathéodory sense, but showing are uniform sense much harder. this note given sequence...

Configurations of type (κ +1)κ give rise to κ–regular simple graphs via configuration graphs. On the other hand, neighbourhood geometries of C4–free κ–regular simple graphs on κ 2 + 1 vertices turn out to be configurations of type (κ + 1)κ. We investigate which configurations of type (κ +1)κ are equal or isomorphic to the neighbourhood geometry of their configuration graph and conversely. We cl...

We call a topological space κ-compact if every subset of size κ has a complete accumulation point in it. Let Φ(μ, κ, λ) denote the following statement: μ < κ < λ = cf(λ) and there is {Sξ : ξ < λ} ⊂ [κ] μ such that |{ξ : |Sξ ∩ A| = μ}| < λ whenever A ∈ [κ]. We show that if Φ(μ, κ, λ) holds and the space X is both μ-compact and λ-compact then X is κ-compact as well. Moreover, from PCF theory we d...

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 ...

The terminology and notation used in this paper are introduced in the following papers: [22], [3], [20], [9], [5], [12], [10], [11], [15], [2], [18], [4], [1], [21], [16], [17], [14], [13], [19], [6], [7], and [8]. For simplicity, we follow the rules: X denotes a complex unitary space, s1, s2, s3 denote sequences of X, R1 denotes a sequence of real numbers, C1, C2, C3 denote complex sequences, ...