Closure operators, irreducibility, Urysohn's lemma, and Tietze extension theorem for proximity spaces

In this paper, we introduce two notions of closure in the category of proximity spaces which satisfy (weak) hereditariness, productivity, and idempotency, and we characterize each of Tᵢ, i=0,1,2, proximity spaces by using these closure operators and show how these subcategories are related. Furthermore, we characterize the irreducible proximity spaces and investigate the relationship among each of irreducible, connected and Tᵢ, i=1,2, proximity spaces. Finally, we present Tietze extension theorem and Urysohn's lemma for proximity spaces.