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Keywords:
merotopic space; nearness space; Cauchy space; filter merotopic space; pretopological space; zero space; complete regularity; weak regularity; $z$-regularity; $\mu$-regularity
merotopic space; nearness space; Cauchy space; filter merotopic space; pretopological space; zero space; complete regularity; weak regularity; $z$-regularity; $\mu$-regularity
Summary:
We conduct an investigation of the relationships which exist between various generalizations of complete regularity in the setting of merotopic spaces, with particular attention to filter spaces such as Cauchy spaces and convergence spaces. Our primary contribution consists in the presentation of several counterexamples establishing the divergence of various such generalizations of complete regularity. We give examples of: (1) a contigual zero space which is not weakly regular and is not a Cauchy space; (2) a separated filter space which is a $z$-regular space but not a nearness space; (3) a separated, Cauchy, zero space which is $z$-regular but not regular; (4) a separated, Cauchy, zero space which is $\mu$-regular but not regular and not $z$-regular; (5) a separated, Cauchy, zero space which is not weakly regular; (6) a topological space which is regular and $\mu$-regular but not $z$-regular; (7) a filter, zero space which is regular and $z$-regular but not completely regular; and, (8) a regular Hausdorff topological space which is $z$-regular but not completely regular.
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