SEPARATION AND SOBER SPACES IN THE CATEGORY OF QUASI-PROXIMITY SPACES

Abstract

The main objective of this paper is to characterize sober spaces, the separation properties T0,T′0,T0, T1 ,PreT2, PreT′2, T2 and T′2 in general in the category of quasi-proximity spaces. Moreover, we introduce two notions of closure in the category of quasi-proximity spaces which satisfy (weak) hereditariness,
productivity, idempotency and we characterize each of Ti, i = 0, 1, 2, quasi-proximity spaces by using these closure operators as well as show how these subcategories are related.