A new kind of pseudo topological (bc) -manifold

This research aims to introduce and explore a new type of manifold space, paving the way for a more detailed study. We begin by defining a new open set (bc~-open) based on filter convergence. Building on this foundation, we define the bc~-chart and bc~-atlas, assuming that the functions involved are bc~-homeomorphisms. The study is then extended to cover functions that are bc~-diffeomorphisms, following a precise definition of this pseudo bc~-smooth manifold. Additionally, we examine functions between multiple manifolds and introduce the concepts of bc~-covering spaces and bc~-covering maps. Finally, we discuss various ideas related to the topic.