Fixed-point theory on neighborhood retracts of convexoid spaces
Given a free ultrafilter p on ℕ we say that x ∈ [0, 1] is the p-limit point of a sequence (x n)n∈ℕ ⊂ [0, 1] (in symbols, x = p -limn∈ℕ x n) if for every neighbourhood V of x, {n ∈ ℕ: x n ∈ V} ∈ p. For a function f: [0, 1] → [0, 1] the function f p: [0, 1] → [0, 1] is defined by f p(x) = p -limn∈ℕ f n(x) for each x ∈ [0, 1]. This map is rarely continuous. In this note we study properties which are equivalent to the continuity of f p. For a filter F we also define the ω F-limit set of f at x. We consider...
We describe the extension of the multiplication on a not-necessarily-discrete topological monoid to its flow compactification. We offer two applications. The first is a nondiscrete version of Hindman’s Theorem, and the second is a characterization of the projective minimal and elementary flows in terms of idempotents of the flow compactification of the monoid.
We show that all finite powers of a Hausdorff space do not contain uncountable weakly separated subspaces iff there is a c.c.c poset such that in is a countable union of -dimensional subspaces of countable weight. We also show that this...
We prove two theorems that characterize tightness in certain products of fans in terms of families of integer-valued functions. We also define several notions of forcing that allow us to manipulate the structure of the set of functions from some cardinal θ to ω, and hence, the tightness of these products. These results give new constructions of first countable <θ-cwH spaces that are not ≤θ-cwH.