APPROXIMATION SPACES VIA IDEALS AND GRILLS.

In this paper, we use the notions of lower set LR(A) and the upper set UR(A) to define the interior operator intAR and the closure operator clAR associated with a set A in an approximation space (X,R). These operators generate an approximation topological space different from the generated Nano topological space in (X,R). Ideal approximation spaces (X,R, ℓ) based on an ideal ℓ joined to the approximation space (X,R) are introduced as well. The approximation continuity and the ideal approximation continuity are defined. The lower separation axioms Ti, i = 0, 1, 2 are introduced in the approximation spaces and also in the ideal approximation spaces. Examples are given to explain the definitions. Connectedness in approximation spaces and ideal connectedness are introduced and the differences between them are explained. The interior and the closure operators are deduced using a grill G defined on (X,R), yielding the same results. [ABSTRACT FROM AUTHOR]