Approach Theory in Merotopic, Cauchy and Convergence Spaces. I

We introduce the notions of approach-merotopic structure and approach-filter merotopic structure by means of a map assigning to a collection of sets 'smallness' of members and define categories AMER and AFIL containing MER and FIL as bireflectively and bicoreflectively embedded subcategories, respectively. We show that the category AMER is a topological construct and AFIL which is a supercategory of ps-MET∞ as well is a cartesian closed topological category bicoreflectively embedded in AMER.