In this paper we apply the technique of measures of noncompactness to the theory of infinite system of integral equations in the Fr´echet spaces. Our aim is to provide a few generalization of Tychonoff fixed point theorem and prove the existence of solutions for infinite systems of nonlinear integral equations with help of the technique of measures of noncompactness and a generalization of Tych...

We study the relationship between the countable axiom of choice and the Tychonoff product theorem for countable families of topological spaces.

Let G be any Hausdorff topological group and let βGX denote the maximal G-compactification of a G-Tychonoff space X (i.e., a Tychonoff G-space possessing a G-compactification). Recall that a completely regular Hausdorff topological space is called pseudocompact if every continuous function f : X →R is bounded. In this paper, we prove that if X and Y are two G-Tychonoff spaces such that the prod...

In their memoir [1, page 54] Alexandroff and Urysohn asked “existe-il un espace compact (bicompact) ne contenant aucun point (κ)?” and went on to remark “La resolution affirmative de ce problème nous donnerait une exemple des espaces compacts (bicompacts) d’une nature toute differente de celle des espaces connus jusqu’à présent”. The ‘compact’ of that memoir is our countably compact, ‘bicompact...

We offer a proof of the duality theorem for finitely presented MV-algebras and rational polyhedra, a folklore and yet fundamental result. Our approach develops first a general dual adjunction between MV-algebras and subspaces of Tychonoff cubes, endowed with the transformations that are definable in the language of MV-algebras. We then show that this dual adjunction restricts to a duality betwe...

We prove that a manifold M is metrisable if and only if its group of homeomorphisms H (M) endowed with the compact-open topology is a qspace. We also discuss pseudo-character and tightness. All spaces under discussion are Tychonoff.