On some generalizations of LC-spaces.
On spaces in which every denumerable subspaces is discrete
On spaces whose nowhere dense subsets are scattered.
On spaces with binary normal subbase
On strongly precontinuous functions.
On supercomplete uniform spaces IV: Countable products
On supercomplete uniform spaces V: Tamano's product problem
On Szymański theorem on hereditary normality of
We discuss the following result of A. Szymański in “Retracts and non-normality points" (2012), Corollary 3.5.: If is a closed subspace of and the -weight of is countable, then every nonisolated point of is a non-normality point of . We obtain stronger results for all types of points, excluding the limits of countable discrete sets considered in “Some non-normal subspaces of the Čech–Stone compactification of a discrete space” (1980) by A. Błaszczyk and A. Szymański. Perhaps our proofs...
On the continuity of the elements of the Ellis semigroup and other properties
We consider discrete dynamical systems whose phase spaces are compact metrizable countable spaces. In the first part of the article, we study some properties that guarantee the continuity of all functions of the corresponding Ellis semigroup. For instance, if every accumulation point of is fixed, we give a necessary and sufficient condition on a point in order that all functions of the Ellis semigroup be continuous at the given point . In the second part, we consider transitive dynamical...
On the existence of P-points in the Stone-Čech compactification of integers
On the existence of true uniform ultrafilters
We shall show that there is an ultrafilter on singular with countable cofinality, which cannot be reached from the set of all subuniform ultrafilters by iterating the closure of sets of size .
On the functionally countable subalgebra of C(X)
On the homology of the Harmonic Archipelago
We calculate the singular homology and Čech cohomology groups of the Harmonic Archipelago. As a corollary, we prove that this space is not homotopy equivalent to the Griffiths space. This is interesting in view of Eda’s proof that the first singular homology groups of these spaces are isomorphic.
On the -Baire property
In this note we show the following theorem: “Let be an almost -discrete space, where is a regular cardinal. Then is -Baire iff it is a -Baire space and every point- open cover of such that is locally- at a dense set of points.” For we obtain a well-known characterization of Baire spaces. The case is also discussed.
On thick spaces
On topological and algebraic structure of extremally disconnected semitopological groups
Starting with a very simple proof of Frol’ık’s theorem on homeomorphisms of extremally disconnected spaces, we show how this theorem implies a well known result of Malychin: that every extremally disconnected topological group contains an open and closed subgroup, consisting of elements of order . We also apply Frol’ık’s theorem to obtain some further theorems on the structure of extremally disconnected topological groups and of semitopological groups with continuous inverse. In particular, every...
On uncountable collections of continua and their span
We prove that if the Euclidean plane contains an uncountable collection of pairwise disjoint copies of a tree-like continuum X, then the symmetric span of X is zero, sX = 0. We also construct a modification of the Oversteegen-Tymchatyn example: for each ε > 0 there exists a tree such that σX < ε but X cannot be covered by any 1-chain. These are partial solutions of some well-known problems in continua theory.
On weakly monotonically monolithic spaces
In this note, we introduce the concept of weakly monotonically monolithic spaces, and show that every weakly monotonically monolithic space is a -space. Thus most known conclusions on -spaces can be obtained by this conclusion. As a corollary, we have that if a regular space is sequential and has a point-countable -network then is a -space.
On -discreteness in uniform spaces
Open mappings on extremally disconnected compact spaces