In the present work, we introduce and examine the framework of fibrewise (fib.wi.) F-and C-topological spaces. In contrast to the standard fib.wi. In topological spaces, the topology in these specific structures is established through a collection of fib.wi. Bundles. The significance of these spaces lies in their flexibility for representing the topological properties of various mathematical structures, including manifolds and group entities. We demonstrate that fib.wi. F-and C-topological spaces are not only straightforward to work with but also offer a rich ground for further mathematical inquiry. This study aims to clarify the underlying definitions of these spaces, their theoretical relevance, and their broader implications in contemporary topology. Specifically, we focus on the construction of novel ideas regarding fib.wi. F-and C-topological spaces defined over a base B. In addition, the paper presents and discusses the properties of fib.wi. F-and C-closed and open spaces over B, along with a number of original propositions.