Ultrametrizable spaces are homeomorphic to clade spaces of pruned trees

TL;DR

This paper shows that all ultrametric spaces can be represented as clade spaces of pruned trees, linking their topological properties to the structure of these trees.

Contribution

It establishes a homeomorphism between ultrametric spaces and clade spaces of pruned trees, providing a new perspective for studying their topological properties.

Findings

Ultrametric spaces are homeomorphic to clade spaces of pruned trees.

Topological properties of ultrametrizable spaces can be characterized via their representing trees.

The approach connects ultrametric topology with ordered tree structures.

Abstract

This paper demonstrates that every ultrametric space is homeomorphic to a clade space of a pruned tree, i.e., a subspace of a tree's canopy. Furthermore, it characterizes several topological properties of ultrametrizable spaces through the features of their representing trees. This approach suggests that topological properties of ultrametrizable spaces should be studies via the study of naturally ordered pruned trees.