A countable connected Urysohn space with a dispersion point that is regular almost everywhere
A countable dense homogeneous set of reals of size ℵ₁
We prove there is a countable dense homogeneous subspace of ℝ of size ℵ₁. The proof involves an absoluteness argument using an extension of the logic obtained by adding predicates for Borel sets.
A countably cellular topological group all of whose countable subsets are closed need not be -factorizable
We construct a Hausdorff topological group such that is a precalibre of (hence, has countable cellularity), all countable subsets of are closed and -embedded in , but is not -factorizable. This solves Problem 8.6.3 from the book “Topological Groups and Related Structures" (2008) in the negative.
A countably compact, separable space which is not absolutely countably compact
We construct a space having the properties in the title, and with the same technique, a countably compact topological group which is not absolutely countably compact.
A counter example on common periodic points of functions.
A counterexample concerning products in the shape category
We exhibit a metric continuum X and a polyhedron P such that the Cartesian product X × P fails to be the product of X and P in the shape category of topological spaces.
A counter-example concerning quasi-homeomorphisms of compacta
A counterexample in non-metric continua theory
A counterexample in semimetric spaces.
A counterexample on a theorem by Khojasteh, Goodarzi, and Razani.
A counter-example on unicoherent Peano spaces
A counterexample to a Fedorenko statement.
A counterexample to a theorem of Argyros
A counterexample to “An extension of Gregus fixed point theorem”.
A counterexample to ”Common fixed point theorem in probabilistic quasi-metric spaces".
A coupled coincidence point theorem in partially ordered metric spaces
A cylinder flow arising from irregularity of distribution
A decomposition theorem for a class of continua for which the set function T is continuous
We prove a decomposition theorem for a class of continua for which F. B.. Jones's set function 𝓣 is continuous. This gives a partial answer to a question of D. Bellamy.
A decomposition theorem for collections of universal subcontinua
A decomposition theorem for compact groups with an application to supercompactness
We show that every compact connected group is the limit of a continuous inverse sequence, in the category of compact groups, where each successor bonding map is either an epimorphism with finite kernel or the projection from a product by a simple compact Lie group. As an application, we present a proof of an unpublished result of Charles Mills from 1978: every compact group is supercompact.