The basis problem for subspaces of monotonically normal compacta

TL;DR

Under Souslin's Hypothesis, the paper shows that uncountable subspaces of zero-dimensional monotonically normal compact spaces always contain an uncountable subset with a familiar topology, revealing a structural property of these spaces.

Contribution

It establishes a new structural result about uncountable subspaces of zero-dimensional monotonically normal compacta assuming Souslin's Hypothesis.

Findings

Uncountable subspaces contain subsets homeomorphic to the real line with standard, Sorgenfrey, or discrete topology.

The result depends on Souslin's Hypothesis, linking set-theoretic assumptions to topological structure.

Provides insight into the internal structure of these compact spaces under specific set-theoretic conditions.

Abstract

We prove, assuming Souslin's Hypothesis, that each uncountable subspace of each zero-dimensional monotonically normal compact space contains an uncountable subset of the real line with either the metric, the Sorgenfrey, or the discrete topology.