We show that a (weakly) Whyburn space may be mapped continuously via an open map onto a non (weakly) Whyburn space . This fact may happen even between topological groups and , a homomorphism, Whyburn and not even weakly Whyburn.
An open continuous map f from a space X onto a paracompact C-space Y admits two disjoint closed sets F₀,F₁ ⊂ X with f(F₀) = Y = f(F₁), provided all fibers of f are infinite and C*-embedded in X. Applications are given to the existence of "disjoint" usco multiselections of set-valued l.s.c. mappings defined on paracompact C-spaces, and to special type of factorizations of open continuous maps from metrizable spaces onto paracompact C-spaces. This settles several open questions.
We construct a completely regular space which is connected, locally connected and countable dense homogeneous but not strongly locally homogeneous. The space has an open subset which has a unique cut-point. We use the construction of a -diffeomorphism of the plane which takes one countable dense set to another.
We prove that the semigroup operation of a topological semigroup extends to a continuous semigroup operation on its Stone-Čech compactification provided is a pseudocompact openly factorizable space, which means that each map to a second countable space can be written as the composition of an open map onto a second countable space and a map . We present a spectral characterization of openly factorizable spaces and establish some properties of such spaces.
In this paper, motivated by questions in Harmonic Analysis, we study the operation of (countable) increasing union, and show it is not idempotent: iterations are needed in general to obtain the closure of a class under this operation. Increasing union is a particular Hausdorff operation, and we present the combinatorial tools which allow to study the power of various Hausdorff operations, and of their iterates. Besides countable increasing union, we study in detail a related Hausdorff operation,...
Using dimension group tools and Bratteli-Vershik representations of minimal Cantor systems we prove that a minimal Cantor system and a Sturmian subshift are topologically conjugate if and only if they are orbit equivalent and Kakutani equivalent.