Part I. On a manifold, we apply the analysis in Part II below to define an intersection called supportive intersection for singular cycles. It has a topological descend to the cup-product. The result is motivated by a problem in cohomology theory. The tool is the notion of currents. A current which is a functional was first introduced by de Rham in 1955. Ever since then, currents played a central role in geometry. However, the part about the support has not been in focus. For instance, the cup-product has been extensively studied in the past. Yet, there is no adequate control on the support of cohomological classes. So, we would like to introduce the supportive intersection that will catch this property. The purpose of this paper is to build the foundation for exploring further.In the end, we'll give an application in this direction.

Part II. This is the technical foundation for the geometry above, but it may have an independent interest. It consists of a functional analysis on a very specific type of convergence of currents. In terms of classical analysis, it is an extension of mollifiers. Classically, mollifier is mostly applied as a smoother for a distribution which is usually viewed as a current of degree $0$. We extend the mollifier to currents where the degrees are positive.