(Non-)contractible road spaces of trees

TL;DR

This paper investigates the topological properties of road spaces of trees, proving contractibility for R-special trees and non-contractibility for certain locally metrizable spaces containing uncountable subspaces.

Contribution

It establishes new results on the contractibility of road spaces of R-special trees and identifies limitations for locally metrizable spaces containing uncountable subspaces.

Findings

Road space of an R-special tree is contractible

Locally metrizable spaces with uncountable ω₁-compact subspaces are not contractible

Raises questions on potential generalizations

Abstract

We prove that the road space of an R-special tree is contractible and that a locally metrizable space containing a copy of an uncountable $\omega_1$-compact subspace of a tree is not. We also raise some questions about possible generalizations.