It is proved that any countable topological group in which the filter of neighborhoods identity element not rapid contains a discrete set with precisely one nonisolated point. This gives negative answer to Protasov's question on existence ZFC nondiscrete all subsets are closed. also extremally disconnected implies ultrafilter and, hence, cannot be constructed ZFC.

We construct in ZFC a countably compact group without non-trivial convergent sequences of size 2c, answering question Bellini, Rodrigues and Tomita [4]. also selectively pseudocompact which is not pracompact, showing that these two properties are equivalent the class topological groups. Using same technique, we has all powers but assuming existence selective ultrafilter.

We give a method for specifying ultrafilter equations and identify their projections on the set of profinite words. Let B be the set of languages captured by first-order sentences using unary predicates for each letter, arbitrary uniform unary numerical predicates and a predicate for the length of a word. We illustrate our methods by giving ultrafilter equations characterising B and then projec...

For a quantale I, which is unit interval endowed with continuous triangular norm and the Barr extension β¯I of ultrafilter monad to I-Rel, characterization discrete presheaf associated given. It also proved that, when & Łucasiewicz norm, isomorphic saturated prefilter monad, product prime functional ideal submonad monad.

We investigate the structure of ultrafilters on Boolean algebras in framework Tukey reducibility. In particular, this paper provides several techniques to construct which are not maximal. Furthermore, we connect analysis with a cardinal invariant algebras, ultrafilter number, and prove consistency results concerning its possible values Cohen random algebras.

There is shown that there exists a complete, atomless, σ-centered Boolean algebra, which does not contain any regular countable subalgebra if and only if there exist a nowhere dense ultrafilter. Therefore the existence of such algebras is undecidable in ZFC. A subalgebra B of a Boolean algebra A is called regular whenever for every X ⊆ B, supB X = 1 implies supA X = 1; see e.g. Heindorf and Sha...

This paper, a continuation of [2]–[8], aims to carry on the project of establishing model-theoretic concepts and methods within the topological context; namely that of compacta (i.e., compact Hausdorff spaces). Since there is a precise duality between the categories of compacta (plus continuous maps) and commutative B-algebras (plus nonexpansive linear maps) (the Gel’fand-Năımark theorem [21]),...

A long-standing question in the theory of measures noncompactness is that for Kuratowski measure α defined on a metric space M, and every bounded subset B⊂M, there countable B0⊂B such α(B0)=α(B)? In this paper, we give an affirmative answer to above. It done by showing each nonempty set B Banach space, so strongly finitely representable B0, free ultrafilter U affinely isometric ultrapower [co(B...

We investigate the possibility of the existence of nonsparse strongly summable ultrafilters on certain abelian groups. In particular, we show that every strongly summable ultrafilter on the countably infinite Boolean group is sparse. This answers a question of Hindman, Steprāns and Strauss.