In this paper, a certain new connectedness of L-fuzzy subsets inL-topological spaces is introduced and studied by means of preclosed sets. Itpreserves some fundamental properties of connected set in general topology.Especially the famous K. Fan’s Theorem holds.

In this paper we introduce the concept of $(1,2)^*$-sb-separated sets and $(1,2)^*$-soft b-connected spaces and prove some properties related to these break topics. Also we disscused the properties of $(1,2)^*$-soft b- compactness in soft bitopological space

Let G $G$ be an affine algebraic group defined over a field k $k$ of characteristic 0. We study the derived moduli space -local systems on pointed connected CW complex X $X$ trivialized at basepoint . This is represented by DG scheme R Loc ( , * ) $ \bm{\mathrm{R}}\mathrm{Loc}_G(X,\ast )$ : we call (co)homology structure sheaf representation homology in and denote it HR \mathrm{HR}_\ast (X,G)$ ...

A general path-lifting theorem, which fails in the context of topological spaces, is shown to hold for toposes, for locales (a slight generalization of topological spaces), and hence for complete separable metric spaces. This result generalizes the known fact that any connected locally connected topos (respectively complete separable metric space) is path-connected. In [MW] we proved that every...