A homogeneous space whose complement is rigid

TL;DR

The paper constructs specific homogeneous subspaces of the Cantor space with dense, rigid complements, advancing understanding of topological rigidity and homogeneity under set-theoretic assumptions.

Contribution

It introduces a method to construct homogeneous subspaces with dense, rigid complements, including under Martin's Axiom, expanding the class of known examples.

Findings

Constructed a homogeneous subspace of $2^\omega$ with a dense, rigid complement.

Under Martin's Axiom, constructed a countable dense homogeneous subspace with a dense, rigid complement.

Demonstrated the existence of such spaces using set-theoretic assumptions.

Abstract

We construct a homogeneous subspace of $2^\omega$ whose complement is dense in $2^\omega$ and rigid. Using the same method, assuming Martin's Axiom, we also construct a countable dense homogeneous subspace of $2^\omega$ whose complement is dense in $2^\omega$ and rigid.