EACH CLOSED SUBSET OF METRIC

ABSTRACT. The main result is an extension theorem (Theorem 1.4) which says that every continuous map, from a closed (•5 subset of an paracompact space X with 0 = Ind X ("Ind " is the large inductive dimension) into a developable space, has a continuous extension to all of X. A corollary is that each closed subset of metric space X with Ind X = 0 is a retract. These results extend work of Robert L. Ellis when the range space is a complete metric space. We also present metrization theorem (Theorem 2.1): every countable compact space with a continuous ultrametric is metrizable. 1. Retracts. A space has large inductive dimension zero or Ind = 0 or is ultranormal if each pair of disjoint closed sets may be separated by a pair of disjoint clopen ( = both open and closed) sets. Nyikos and Reichel noted in [6] that ultranormal spaces are important in the theories for both the Stone duality for Boolean Algebras and the p-adic topologies. Staum has shown in [7] that all the Gelfand topologies (on the families of maximal ideal of a commutative non-archimedean Banach algebra over a field with identity of norm 1) are ultraregular. Much on the structure of ultranormal spaces is presented in [2], [3] and [5]- [7]. Ellis' paper [2] provides,,fiqcient background for reading this paper. This section is on retractions of and continuous extensions of maps into ultranormal spaces. Section 2 presents ametrization theorem for spaces with a "continuous ultrametric." Key wo•el • aneI phra•e•. retract, extension, zero-dimensional, ultraparacompact, ultra-metric, metfizable. * This is a rewriting of my earlier report [1