The aim of this paper is to study fuzzy extensions some covering properties defined by A. V. Arhangel’skii and studied other authors. Indeed, in 2016, paracompact-type properties: C-paracompactness C2-paracompactness. Later, M. Saeed, L. Kalantan H. Alzumi investigated these two properties. In paper, we define notions obtain results about them, particular, prove that are good those Arhangel’skii.

We prove some properties of several types of connectedness defined in intuitionistic fuzzy special topological spaces.

In this paper we obtain some properties of the weak smooth α-closure and weak smooth α-interior of a fuzzy set in smooth topological spaces and introduce the concepts of several types of weak∗ smooth compactness in smooth topological spaces and investigate some of their properties.