Combinatorial aspects of measure and category
Combinatorial lemmas for polyhedrons
We formulate general boundary conditions for a labelling to assure the existence of a balanced n-simplex in a triangulated polyhedron. Furthermore we prove a Knaster-Kuratowski-Mazurkiewicz type theorem for polyhedrons and generalize some theorems of Ichiishi and Idzik. We also formulate a necessary condition for a continuous function defined on a polyhedron to be an onto function.
Combinatorial lemmas for polyhedrons I
We formulate general boundary conditions for a labelling of vertices of a triangulation of a polyhedron by vectors to assure the existence of a balanced simplex. The condition is not for each vertex separately, but for a set of vertices of each boundary simplex. This allows us to formulate a theorem, which is more general than the Sperner lemma and theorems of Shapley; Idzik and Junosza-Szaniawski; van der Laan, Talman and Yang. A generalization of the Poincaré-Miranda theorem is also derived.
Combinatorial properties of uniformities
Combinatorial trees in Priestley spaces
We show that prohibiting a combinatorial tree in the Priestley duals determines an axiomatizable class of distributive lattices. On the other hand, prohibiting -crowns with does not. Given what is known about the diamond, this is another strong indication that this fact characterizes combinatorial trees. We also discuss varieties of 2-Heyting algebras in this context.
Combinatorics of ideals --- selectivity versus density
This note is devoted to combinatorial properties of ideals on the set of natural numbers. By a result of Mathias, two such properties, selectivity and density, in the case of definable ideals, exclude each other. The purpose of this note is to measure the ``distance'' between them with the help of ultrafilter topologies of Louveau.
Combinatorics of open covers (III): games, Cp (X)
Some of the covering properties of spaces as defined in Parts I and II are here characterized by games. These results, applied to function spaces of countable tightness, give new characterizations of countable fan tightness and countable strong fan tightness. In particular, each of these properties is characterized by a Ramseyan theorem.
Combinatorics of open covers (VII): Groupability
We use Ramseyan partition relations to characterize: ∙ the classical covering property of Hurewicz; ∙ the covering property of Gerlits and Nagy; ∙ the combinatorial cardinal numbers and add(ℳ ). Let X be a -space. In [9] we showed that has countable strong fan tightness as well as the Reznichenko property if, and only if, all finite powers of X have the Gerlits-Nagy covering property. Now we show that the following are equivalent: 1. has countable fan tightness and the Reznichenko property. 2....
Commemoration of Eduard Čech
Commemoration of Eduard Čech
Commemoration of Eduard Čech
Comments concerning flows near compact invariant sets
Comments on a paper of Pachpatte
Comments on nonstandard topology
Comments on the definition of a Tikhonov semifield homomorphism
Comments on the paper by Gordji and Baghani entitled “A generalization of Nadler's fixed point theorem".
Comments on the papers “Arch Math. (Brno), 42 (2006), 51–58”, “Thai J. Math., 3 (2005), 63–70” and “Math. Communications 13 (2008), 85–96”.
Common coincidence points of -weakly commuting maps.
Common coupled fixed point theorems for contractive mappings in fuzzy metric spaces.
Common fixed point and approximation results for noncommuting maps on locally convex spaces.