In this paper we introduce a new definition of the first non-abelian cohomology of topological groups.  We relate the cohomology of a normal subgroup $N$ of a topological group $G$ and the quotient $G/N$ to the cohomology of $G$. We get the inflation-restriction exact sequence. Also, we obtain a seven-term exact cohomology sequence up to dimension 2. We give an interpretation of the first non-a...

Let $ { mathcal{R}} L$ be the ring of real-valued continuous functions on a frame $L$ as the pointfree  version of $C(X)$, the ring of all real-valued continuous functions on a topological space $X$. Since $C_c(X)$ is the largest subring of $C(X)$ whose elements have countable image, this motivates us to present the pointfree  version of $C_c(X).$The main aim of this paper is to present t...

In this paper we characterize various sorts of boundedness of the free (abelian) topological group F (X) (A(X)) as well as the free locally-convex linear topological space L(X) in terms of properties of a Tychonoff space X. These properties appear to be close to so-called selection principles, which permits us to show, that (it is consistent with ZFC that) the property of Hurewicz (Menger) is l...

Let L ◦ be a propositional language with standard Boolean connectives plus two modalities: an S4-ish topological modality and a temporal modality ◦, understood as ‘next’. We extend the topological semantic for S4 to a semantics for the language L ◦ by interpreting L ◦ in dynamic topological systems, i.e., ordered pairs 〈X, f 〉, where X is a topological space and f is a continuous function on X ...

A stratified bundle is a fibered space in which strata are classical bundles and in which attachment of strata is controlled by a structure category F of fibers. Well known results on fibre bundles are shown to be true for stratified bundles; namely the pull back theorem, the bundle theorem and the principal bundle theorem. AMS SC : 55R55 (Fiberings with singularities); 55R65 (Generalizations o...

In 1997, Fang proposed the concept of boundedness of $L$-fuzzy setsin $L$-topological vector spaces. Since then, this concept has beenwidely accepted and adopted in the literature. In this paper,several characterizations of bounded $L$-fuzzy sets in$L$-topological vector spaces are obtained and some properties ofbounded $L$-fuzzy sets are investigated.

A stratified bundle is a fibered space in which strata are classical bundles and in which attachment of strata is controlled by a structure category F of fibers. Well known results on fibre bundles are shown to be true for stratified bundles; namely the pull back theorem, the bundle theorem and the principal bundle theorem. AMS SC : 55R55 (Fiberings with singularities); 55R65 (Generalizations o...

We develop the homotopy theory of Euler characteristic (magnitude) of a category enriched in a monoidal model category. If a monoidal model category V is equipped with an Euler characteristic that is compatible with weak equivalences and fibrations in V, then our Euler characteristic of V-enriched categories is also compatible with weak equivalences and fibrations in the canonical model structu...

in this paper, the notion of fuzzy semicompactness degrees isintroduced in $l$-fuzzy topological spaces by means of theimplication operation of $l$. characterizations of fuzzysemicompactness degrees in $l$-fuzzy topological spaces  areobtained, and some properties of fuzzy semicompactness degrees areresearched.

Powar P. L. and Rajak K., have introduced fine-topological space which is a special case of generalized topological sspace. In this paper, the author has introduced a new class of functions called fine weakly BR-continuous functions on fine topological space (cf. [12]). Weakly BR-continuous functions has been introduced by Ekici [3] which includes the class of strongly continuous functions due ...