Vol. 104, No. 1, 1983

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Trees and proto-metrizable spaces

Luther Bush Fuller

Vol. 104 (1983), No. 1, 55–75

It is known that metrizable spaces are characterized as the compact closed continuous image of a subspace of some Baire zero-dimensional space and that compact metrizable spaces are characterized as the closed continuous image of the Cantor set.

In this paper we investigate some of the properties of trees and non-archemedian spaces and provide, among others, a characterization of proto-metrizable spaces which generalizes the above characterizations of metrizable spaces by showing that a proto-metrizable space is the image of a non-archemedian space under an (irreducible) closed map such that each point pre-image is either a point or a compact Gδ-set This result specializes a characterization of paracompact spaces as the image under a compact closed map of an ultra-paracompact space.

We then show that non-archemedian spaces and their irreducible closed continuous images have a normality property, called M1-normality, and it follows that proto-metrizable spaces are M1-normal.

Mathematical Subject Classification 2000
Primary: 54E99
Secondary: 54C99, 54D15
Received: 14 February 1978
Revised: 26 June 1980
Published: 1 January 1983
Luther Bush Fuller