Separation and sober spaces in the category of quasi-proximity spaces

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 operators 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.