Remove compactness from the tree-space fixed-point theorem

Determine whether compactness can be omitted from the fixed-point theorem asserting that every continuous feebly+ topological contraction on a tree space has a unique fixed point.

Background

The paper defines tree spaces using a tree of nested point-classes whose infinite branches have singleton intersection, and proves that every continuous feebly+ contraction on a compact tree space has a unique fixed point.

The proof uses compactness to ensure that a decreasing sequence of nonempty closed invariant sets has nonempty intersection. It remains unresolved whether the same fixed-point conclusion holds for noncompact tree spaces, or whether an alternative weaker hypothesis can replace compactness.