A note on proto-metrisable spaces
Protometrizable spaces were introduced by \textit{P. J. Nyikos} [Stud. Topol., Conf. Charlotte, N.C., 1974, 427-450 (1975; Zbl 0337.54014)] and have subsequently been studied by a number of authors, especially by \textit{G. Gruenhage} and \textit{P. Zenor} [Houston J. Math. 3, 47-53 (1977; Zbl 0346.54012)]. The proto-typical protometrizable space is the Michael line [\textit{E. Michael}, Bull. Am. Math. Soc. 69, 375-376 (1963; Zbl 0114.389)]. The principal results are new characterizations of protometrizable spaces, some of a `monotonic' manner, and one in terms of the `structuring mechanism' introduced by the reviewer and \textit{A. W. Roscoe} [Proc. Am. Math. Soc. 90, 631-640 (1984; Zbl 0541.54034)]. Fuller use of the structure provided by the subset relation has been long overdue in topology. Monotone normality and stratifiability (which can be viewed as `monotone perfect normality') are monotone properties taking advantage of this structure. This paper demonstrates the fruitfulness of the approach. The work has been extended by \textit{H. J. K. Junnila} and \textit{H.-P. A. Künzi} [Ortho-bases and monotonic properties, Proc. Am. Math. Soc. 119, No. 4, 1335-1345 (1993)]. The authors give an application of their results to topological groups, which has an interesting corollary: a `small' (for example, separable or of countable pseudo-character) topological group which is `translation invariantly monotonically normal' is metrizable. It is immediate from the structuring mechanism characterization of protometrizable spaces that such spaces are well ordered (F) spaces [the authors, Topol. Proc. 17, 111-130 (1992; see Zbl 0797.54038 below)].
scientific article; zbMATH DE number 447262
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