Non-trivial copies of N*

TL;DR

This paper demonstrates the consistency of having complex, non-trivial subspaces and self-maps within the Čech–Stone remainder of the natural numbers, despite all autohomeomorphisms being trivial.

Contribution

It establishes the possibility of non-trivial copies and self-maps of * under certain set-theoretic assumptions, challenging previous intuitions.

Findings

Existence of regular closed non-clopen copies of *

Existence of non-trivial self-maps of *

All autohomeomorphisms of * can be trivial under certain models

Abstract

We show that it is consistent to have regular closed non-clopen copies of $\mathbb N^*$ within $\mathbb N^*$ and a non-trivial self-map of $\mathbb N^*$ even if all autohomeomorphisms of $\mathbb N^*$ are trivial.