The category of pointwise S-proximity spaces

First it is proved that the category of pointwise quasi-uniform spaces and pointwise quasi-uniformly continuous morphisms is topological. Then the concept of pointwise S-quasi-proximity on lattice-valued fuzzy set theory is introduced in such a way as to be compatible with pointwise quasi-uniformity and pointwise p.q. metric. Each co-topology can be induced by a pointwise S-quasi-proximity. When a valued lattice is a completely distributive lattices equipped with an order-reversing involution, the concept of pointwise S-proximity can be defined by means of pointwise S-quasi-proximity. The relation between pointwise S-(quasi-)proximities and pointwise (quasi-)uniformities is discussed. It is proved that the category of pointwise S-quasi-proximity spaces is isomorphic to the category of totally bounded pointwise quasi-uniform spaces. The category of pointwise S-proximity spaces is isomorphic to a subcategory of totally bounded pointwise uniform spaces. Hence they are topological.