In this paper, an explicit characterization of the separation properties T0 and T1 at a point p is given in the topological category of proximity spaces. Furthermore, the (strongly) closed and (strongly) open subobjects of an object are characterized in the category of proximity spaces and also the characterization of each of the various notions of the connected objects in this category are given.

In this paper, an explicit characterization of the separation properties for T0, T1, PreT2 (pre-Hausdorff) and T2 (Hausdorff) is given in the topological category of proximity spaces. Moreover, specific relationships that arise among the various Ti,i = 0,1,2 and PreT2 structures are examined in this category. Finally, we investigate the relationships among generalized separation properties for Ti,i = 0,1,2 and PreT2 (in our sense), separation properties at a point p and separation properties for Ti, i = 0,1,2 in the usual sense in this category.

In this paper, we characterize local T0 and T1 quantale-valued gauge spaces, show how these concepts are related to each other and apply them to L-approach distance spaces and L-approach system spaces. Furthermore, we give the characterization of a closed point and D-connectedness in quantale-valued gauge spaces. Finally, we compare all these concepts to each other.

We characterize the separation properties T0 and T1 at a point p in the category of quasi-proximity spaces. Moreover, the (strongly) closed and (strongly) open subobjects of an object, and each of the various notions of connected and compact objects are characterized in this topological category.

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