Defining New Structures on a Universal Set: Diving Structures and Floating Structures

This paper introduces new structures (DV, FL) on a set Ξ. They will be known as diving structures and floating structures. The pairs (Ξ,DV), (Ξ,FL) will be called diving spaces and floating spaces, respectively. These structures are not related to each other and are not related to any of the previous common structures such as topology, ideal, filter, grill or primal. We study some properties to characterize these new notions. Separation axioms and connectedness will be defined in these new spaces. When R is an equivalence relation on Ξ, rough sets in diving spaces and rough sets in floating spaces will also be defined. Thus, (Ξ,R,DV) and (Ξ,R,FL) are called the diving approximation spaces and the floating approximation spaces, respectively.