The aim this paper is to present an answer to Problem 1 of Monk [10], [11]. We do this by proving in particular that if μ is a strong limit singular cardinal, and then there are Boolean algebras such that . Further we improve this result, deal with the method and the necessity of the assumptions. In particular we prove that if is a ccc Boolean algebra and then satisfies the λ-Knaster condition (using the “revised GCH theorem”).
We discuss various generalizations of the class of Lindelöf spaces and study the difference between two of these generalizations, the classes of star-Lindelöf and centered-Lindelöf spaces.
A bottleneck in a dendroid is a continuum that intersects every arc connecting two non-empty open sets. Piotr Minc proved that every dendroid contains a point, which we call a center, contained in arbitrarily small bottlenecks. We study the effect that the set of centers in a dendroid has on its structure. We find that the set of centers is arc connected, that a dendroid with only one center has uncountably many arc components in the complement of the center, and that, in this case, every open set...
A subset of a metric space is central iff for every Katětov map upper bounded by the diameter of and any finite subset of there is such that for each . Central subsets of the Urysohn universal space (see introduction) are studied. It is proved that a metric space is isometrically embeddable into as a central set iff has the collinearity property. The Katětov maps of the real line are characterized.
Let K(2ℕ) be the class of compact subsets of the Cantor space 2ℕ, furnished with the Hausdorff metric. Let f ∈ C(2ℕ). We study the map ω f: 2ℕ → K(2ℕ) defined as ω f (x) = ω(x, f), the ω-limit set of x under f. Unlike the case of n-dimensional manifolds, n ≥ 1, we show that ω f is continuous for the generic self-map f of the Cantor space, even though the set of functions for which ω f is everywhere discontinuous on a subsystem is dense in C(2ℕ). The relationships between the continuity of ω f and...
A homeomorphism f : X → X of a compactum X is expansive (resp. continuum-wise expansive) if there is c > 0 such that if x, y ∈ X and x ≠ y (resp. if A is a nondegenerate subcontinuum of X), then there is n ∈ ℤ such that (resp. ). We prove the following theorem: If f is a continuum-wise expansive homeomorphism of a compactum X and the covering dimension of X is positive (dim X > 0), then there exists a σ-chaotic continuum Z = Z(σ) of f (σ = s or σ = u), i.e. Z is a nondegenerate subcontinuum...