Seperation, Irreducibility, Urysohn's Lemma and Tietze Extention Theorem for Cauchy Spaces

Abstract

In this paper, we introduce two notions of closure in the category of Cauchy  spaces  which satisfy (weakly) hereditariness, productivity and idempotency, and we characterize each of $T_{i}, i=0,1,2$ cauchy spaces  by using these closure operators as well as show each each of these subcategories are isomorphic. Furthermore, we characterize the irreducible Cauchy  spaces  and examine the relationship among each of irreducible, connected Cauchy  spaces. Finally, we present Urysohn’s lemma and Tietze extension theorem for Cauchy  spaces.