In this paper, some fundamental concepts are given relating to fuzzytopological spaces. Then it is shown that there is a contravariant functorfrom the category of the pointed fuzzy topological spaces to the category ofgroups and homomorphisms. Also the fuzzy topological spaces which are Hopfspaces are investigated and it is shown that a pointed fuzzy toplogicalspace having the same homotopy typ...

Consider a connected topological space $$X$$ with point $$x$$ in and let $$K$$ be field the discrete topology. We study Tannakian category of finite-dimensional (flat) vector bundles on its dual $$\pi(X,x)$$ respect to fiber functor . The maximal pro-étale quotient is étale fundamental group studied by Kucharczyk Scholze. For well-behaved spaces, pro-algebraic completion ordinary group. obtain ...

In this paper, we define topological hyperrings and study their basic concepts which supported by illustrative examples. We show some differences between topological rings and topological hyperrings. Also, by the fundamental relation $\Gamma^{*}$, we indicate the role of complete parts (saturated subsets) and complete hyperrings in topological hyperrings and specially we show that if every (clo...

For a locally path connected topological space, the topological fundamental group is discrete if and only if the space is semilocally simplyconnected. While functoriality of the topological fundamental group for arbitrary topological spaces remains an open question, the topological fundamental group is always a homogeneous space.

The concept of a topology for the fundamental group appears to have originated with Hurewicz (1935), but it has been actively studied only for the last ten years. The classical fundamental group of a pointed topological space X naturally inherits the quotient topology from the space of based loops in X endowed with the compact-open topology. The resulting space, called the topological fundament...

we introduce some new concepts of topological spaces which say α − separable topologicalspace and o-topological group, α − first axiom, α − second axiom, and we find some relations betweenthem with some applications in normed spaces.

In this paper, using a relation between Schur multipliers of pairs and triples of groups, the fundamental group and homology groups of a homotopy pushout of Eilenberg-MacLane spaces, we present among other things some behaviors of Schur multipliers of pairs and triples with respect to free, amalgamated free, and direct products and also direct limits of groups with topological approach.

Abstract. In this paper, we first derive specific results concerning the continuity and upper semi-continuity of the spectral radius and spectrum functions on fundamental locally multiplicative topological algebras. We continue our investigation by further determining the automatic continuity of linear mappings and homomorphisms in these algebras.

in this paper, we introduce and study the notion of precompacttopological generalized groups and some new results are given.