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

Authors: SAMED ÖZKAN, MUAMMER KULA, SÜMEYYE KULA, TESNİM MERYEM BARAN

Abstract: 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 $T_{i}, i=0,1,2$, 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 $T_{i}, i=1,2$, proximity spaces. Finally, we present Tietze extension theorem and Urysohn's lemma for proximity spaces.

Keywords: Topological category, proximity space, closure operators, irreducible objects

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