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.
