A systematic analysis is made of the character of the free and free abelian topological groups on uniform spaces and on topological spaces. In the case of the free abelian topological group on a uniform space, expressions are given for the character in terms of simple cardinal invariants of the family of uniformly continuous pseudometrics of the given uniform space and of the uniformity itself....

A common theme in algebraic topology is to study functors from the homotopy category of topological spaces to various categories of algebraic objects, such as the classical homology functor, the cohomology functor and the homopoty functor. In this talk, I will introduce some functors from topological spaces to Lie algebras and algebraic varieties, and explore their properties. We can use these ...

Every nontrivial countably productive coreflective subcategory of topological linear spaces is κ-productive for a large cardinal κ (see [10]). Unlike that case, in uniform spaces for every infinite regular cardinal κ, there are coreflective subcategories that are κ-productive and not κ-productive (see [8]). From certain points of view, the category of topological groups lies in between those ca...

Having as starting point Barr’s description of topological spaces as lax algebras for the ultrafilter monad [2], in this paper we present further topological examples of lax algebras – such as quasi-metric spaces, approach spaces and quasi-uniform spaces – and show that, in a suitable setting, the categories of lax algebras have indeed a topological nature. Furthermore, we generalize to this se...

In this article on fuzzy topological spaces we studied and the further investigated different kind of continues mappings in fuzzy topological space. In this article fuzzy sets have been defined and use of fuzzy numbers in daily life. In this article we have investigated how the fuzzy topological spaces have been defined by different authors and what is its future scope.

In this paper, the notion of almost S^{*}-compactness in L-topologicalspaces is introduced following Shi’s definition of S^{*}-compactness. The propertiesof this notion are studied and the relationship between it and otherdefinitions of almost compactness are discussed. Several characterizations ofalmost S^{*}-compactness are also presented.

The paper is concerned with order-topological characterizations of topological Riesz spaces, in particular spaces of measurable functions, not containing Riesz isomorphic or linearly homeomorphic copies of c0 or `∞.

We present a topological minimax theorem (Theorem 2.2). The topological assumptions on the spaces involved are somewhat weaker than those usually found in the literature. Even when reinterpreted in the convex setting of topological vector spaces, our theorem yields nonnegligible improvements, for example, of the Passy–Prisman theorem and consequently of the Sion theorem, contrary to most result...