H. L. Bentley, E. Lowen-Colebunders Completely regular spaces
Comment.Math.Univ.Carolinae 32,1 (1991) 129-154.
Abstract: 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 counterex- amples establishing the divergence of various such generalizations of complete reg- ularity. 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 az-regular space but not a nearness space; (3) a separated, Cauchy, zero space which isz-regular but not regular; (4) a separated, Cauchy, zero space which isµ-regular but not regular and notz-regular; (5) a separated, Cauchy, zero space which is not weakly regular;
(6) a topological space which is regular andµ-regular but notz-regular; (7) a filter, zero space which is regular and z-regular but not completely regular; and, (8) a regular Hausdorff topological space which isz-regular but not completely regular.
Keywords: merotopic space, nearness space, Cauchy space, filter merotopic space, pretopological space, zero space, complete regularity, weak regularity,z-regularity, µ-regularity
AMS Subject Classification: 54C30, 54C40, 54E17, 18B30
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