On the Cantor and Hilbert Cube Frames and the Alexandroff-Hausdorff Theorem

Abstract

The aim of this work is to give a pointfree description of the Cantor set. It can be shown that the Cantor set is homeomorphic to the $p$-adic integers $\mathbb{Z}_{p}:=\{x\in\mathbb{Q}_{p}: |x|_p\leq 1\}$ for every prime number $p$. To give a pointfree description of the Cantor set, we specify the frame of $\mathbb{Z}_{p}$ by generators and relations. We use the fact that the open balls centered at integers generate the open subsets of $\mathbb{Z}_{p}$ and thus we think of them as the basic generators; on this poset we impose some relations and then the resulting quotient is the frame of the Cantor set $\mathcal{L}(\mathbb{Z}_{p})$. We prove that $\mathcal{L}(\mathbb{Z}_{p})$ is a spatial frame whose space of points is homeomorphic to $\mathbb{Z}_{p}$. In particular, we show with pointfree arguments that $\mathcal{L}(\mathbb{Z}_{p})$ is $0$-dimensional, (completely) regular, compact, and metrizable (it admits a countably generated uniformity). Finally, we give a point-free counterpart of the Hausdorff-Alexandroff Theorem which states that \emph{every compact metric space is a continuous image of the Cantor space} (see, e.g. \cite{Alexandroff} and \cite{Hausdorff}). We prove the point-free analog: if $L$ is a compact metrizable frame, then there is an injective frame homomorphism from $L$ into $\mathcal{L}(\mathbb{Z}_{2})$.