Seperation, irreducibility, Urysohn?s lemma and Tietze extension theorem for Cauchy spaces

In this paper, we introduce two notions of closure operators in the category of Cauchy spaces which satisfy (weak) hereditariness, productivity and idempotency, and we characterize each of Ti, i = 0,1, 2 cauchy spaces by using these closure operators as well as show 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.

Automated summary

In this paper, we introduce two notions of closure operators in the category of Cauchy spaces, satisfying (weak) hereditariness, productivity and idempotency. We characterize each of Ti, i = 0,1, 2 cauchy spaces using these closure operators, and show that each of these subcategories are isomorphic. Furthermore, we examine the relationship among irreducible and connected Cauchy spaces, and present Urysohn's lemma and Tietze extension theorem for Cauchy spaces.