A locally connected topos is a Galois topos if the Galois objects generate the topos. We show that the full subcategory of Galois objects in any connected locally connected topos is an inversely 2-filtered 2-category, and as an application of the construction of 2-filtered bilimits of topoi, we show that every Galois topos has a point. introduction. Galois topoi (definition 1.5) arise in Grothe...

In this paper, we study soft sets of the first and second Baire categories. The category are examined to be small from point view topology, while large. family in a topological space forms σ-ideal. This contributes development theory ideal topology. main properties these classes discussed. concepts points where or introduced. These types subclasses non-cluster cluster sets. Then, various result...

A kind of unstable homotopy theory on the category of associative rings (without unit) is developed. There are the notions of fibrations, homotopy (in the sense of Karoubi), path spaces, Puppe sequences, etc. One introduces the notion of a quasi-isomorphism (or weak equivalence) for rings and shows that similar to spaces the derived category obtained by inverting the quasiisomorphisms is natura...

In this paper, we first present a new type of the concept of open sets by expressing some properties of arbitrary mappings on a power set. With the generalization of the closure spaces in categorical topology, we introduce the generalized topological spaces and the concept of generalized continuity and become familiar with weak and strong structures for generalized topological spaces. Then, int...

We show that, in any Mal’tsev (and a fortiori protomodular) category E, not only the fibre GrdXE of internal groupoids above the object X is a naturally Mal’tsev category, but moreover it shares with the category Ab of abelian groups the property following which the domain of any split epimorphism is isomorphic with the direct sum of its codomain with its kernel. This allows us to point at a ne...