Extension of uniformly disconnected metrics and ultrametrics
by
Ihor Stasyuk
Lviv National University, Lviv, Ukraine, University of Saskatchewan, Canada
Coauthors: E.D. Tymchatyn

A metric r on a set Y is called an ultrametric if r(x, y) ≤ max{r(x, y), r(y, z)} for all x, y, z ∈ Y. A metric r on Y is called uniformly disconnected if there is c > 0 such that r(x_i-1, x_i) < cr(x₁, x_n), i ∈ {1, ..., n} for every finite subset {x₀, ..., x_n} of Y. We construct an extension operator for uniformly disconnected metrics defined on the family of closed subsets of a compact zero-dimensional metric space. We also construct an operator which simultaneously extends continuous ultrametrics defined on closed subsets of any complete ultrametric space. These two operators possess several natural properties such as continuity and positive-homogeneity. They also preserve norms, maxima of metrics with common domains and the Assouad dimension of a metric space. These results generalize Tymchatyn-Zarichnyi theorem on extending ultrametrics for a compact space.

Date received: February 18, 2008