Continuous Maps on Aronszajn Trees

TL;DR

This paper investigates the existence of continuous order-preserving maps from special Aronszajn trees into totally imperfect sets of real numbers under Jensen's diamond principle, revealing limitations on such mappings.

Contribution

It demonstrates that under Jensen's diamond, certain Aronszajn trees cannot be continuously mapped into totally imperfect sets, highlighting new constraints in set-theoretic topology.

Findings

No continuous order-preserving map exists into B under the given assumptions.

The result depends on Jensen's diamond principle.

It advances understanding of mappings between complex trees and real sets.

Abstract

Assuming Jenson's principle diamond: Whenever B is a totally imperfect set of real numbers, there is special Aronszajn tree with no continuous order preserving map into B.