Abstract
In this Chapter, we introduced the interior and the closure operators with respect to an ideal defined on an approximation space generating an ideal approximation space. The approximation spaces have no relation to the associated Nano topological space. Separation axioms and connectedness in approximation spaces and in ideal approximation spaces are defined and studied. Also, we defined grill separation axioms and grill approximation connectedness with respect to a given grill. All results with ideal are the same with respect to a grill and the converse is true. There is an agreement between the notions ideal and grill. Some examples are given to confirm the introduced implications.
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