${\mathbb{M}}^{\ast }$, ${\mathbb{N}}^{\ast }$, and ${\mathbb{H}}^{\ast }$

Abstract

Let $\mathbb{M}=\mathbb{N}\times [0,1]$. The natural projection $\ddot{I}€:\mathbb{M}\to \mathbb{N}$, which sends $(n,x)$ to $n$, induces a projection mapping $\ddot{I}{€}^{\ast }:{\mathbb{M}}^{\ast }\to {\mathbb{N}}^{\ast }$, where ${\mathbb{M}}^{\ast }$ and ${\mathbb{N}}^{\ast }$ denote the Čech-Stone remainders of $\mathbb{M}$ and $\mathbb{N}$, respectively. We show that $\mathsf{CH}$ implies every autohomeomorphism of ${\mathbb{N}}^{\ast }$ lifts through the natural projection to an autohomeomorphism of ${\mathbb{M}}^{\ast }$. That is, for every homeomorphism $h:{\mathbb{N}}^{\ast }\to {\mathbb{N}}^{\ast }$ there is a homeomorphism $H:{\mathbb{M}}^{\ast }\to {\mathbb{M}}^{\ast }$ such that $\ddot{I}{€}^{\ast }\circ H=h\circ \ddot{I}{€}^{\ast }$. This complements a recent result of the second author, who showed that this lifting property is not a consequence of $\mathsf{ZFC}$. Combining this lifting theorem with a recent result of the first author, we also prove that $\mathsf{CH}$ implies there is an order-reversing autohomeomorphism of~${\mathbb{H}}^{\ast }$, the Čech-Stone remainder of the half line $\mathbb{H}=[0,\infty )$.