关于Cantor和Hilbert立方框架及Alexandroff-Hausdorff定理
On the Cantor and Hilbert Cube Frames and the Alexandroff-Hausdorff Theorem
摘要：
本文的目的是给出Cantor集的一个无点描述。可以证明，对于每个素数$p$，康托集同胚于$p$ -进整数$\mathbb{Z}_{p}:=\{x\in\mathbb{Q}_{p}: |x|_p\leq 1\}$.为了给出Cantor集的无点描述，我们用生成元和关系指定了$\mathbb{Z}_{p}$的框架.我们利用以整数为中心的开球生成$\mathbb{Z}_{p}$的开子集的事实，因此我们认为它们是基本生成元;在这个偏序集上，我们施加一些关系，然后得到的商是康托集$\mathcal{L}(\mathbb{Z}_{p})$的框架。我们证明了$\mathcal{L}(\mathbb{Z}_{p})$是一个空间标架，它的点空间同胚于$\mathbb{Z}_{p}$。特别是，我们证明了$\mathcal{L}(\mathbb{Z}_{p})$是$0$ -维，（完全）定期，紧凑，和可度量化（它承认一个可数生成的均匀性）与点无关的参数。最后，我们给出了Hausdorff-Alexandroff定理的一个无点对应定理，该定理指出\emph{every compact metric space is a continuous image of the Cantor space}（例如，参见\cite{Alexandroff}和\cite{Hausdorff}）。我们证明了无点类比：如果$L$是一个紧可度量化框架，则存在一个从$L$到$\mathcal{L}(\mathbb{Z}_{2})$的内射框架同态。
英文摘要：
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 $\mathcal{L}(\mathbb{Z}_{2})$.
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