Continuous Function Compact Set Image Complete

On the Cantor and Hilbert cube frames and the Alexandroff-Hausdorff theorem

Abstract

The aim of this work is to give a point-free description of the Cantor set. It can be shown (see, e.g. [18]) that the Cantor set is homeomorphic to the p-adic integers ${\mathbb{Z}}_p:=\{x\in {\mathbb{Q}}_p:|x{|}_p\le 1\}$ for every prime number p. To give a point-free 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 $\mathcal{L}({\mathbb{Z}}_p)$ .

A topological characterization of it is given by Brouwer's Theorem [7]: The Cantor set is the unique totally disconnected, compact metric space with no isolated points. 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 point-free arguments that $\mathcal{L}({\mathbb{Z}}_p)$ is 0-dimensional, (completely) regular, compact, and metrizable (it admits a countably generated uniformity). Moreover, we show that the frame $\mathcal{L}({\mathbb{Z}}_p)$ satisfies $cb{d}_{\mathcal{L}({\mathbb{Z}}_p)}(0)=0$ , where $cb{d}_{\mathcal{L}({\mathbb{Z}}_p)}:\mathcal{L}({\mathbb{Z}}_p)\to \mathcal{L}({\mathbb{Z}}_p)$ , defined by $cb{d}_{\mathcal{L}({\mathbb{Z}}_p)}(a)=\bigwedge \{x\in \mathcal{L}({\mathbb{Z}}_p)|x\le a\text{ and}(x\to a)=a\}$ is the Cantor-Bendixson derivative (see, e.g. [19]). It follows that a frame L is isomorphic to $\mathcal{L}({\mathbb{Z}}_p)$ if and only if L is a 0-dimensional compact regular metrizable frame with $cb{d}_L(0)=0$ . Finally, we give a point-free counterpart of the Hausdorff-Alexandroff Theorem which states that every compact metric space is a continuous image of the Cantor space (see, e.g. [1] and [12]). We prove the point-free analogue: if L is a compact metrizable frame, then there is an injective frame homomorphism from L into $\mathcal{L}({\mathbb{Z}}_2)$ .

Introduction

Frames (locales, complete Heyting algebras) appeared as an algebraic manifestation of topological spaces. Indeed, for any topological space S its topology $\mathrm{\Omega }(S)$ is a frame. The category Frm consists of frames as objects and $\bigvee ,\wedge$ -preserving functions as morphisms, we have a contravariant adjunction (see [13] or [16] for details): Several point-sensitive constructions have a frame theoretical counterpart. For example, Joyal in [14] produces a point-free construction of the reals where a detailed construction can be found in [13, Chapter IV]. Banaschewski in [5] gives another construction of the frame of the reals using generators and relations where aspects of the reals and its rings of continuous functions can be translated into this setting (see [16, Chapter XIV]). Later, in [4] the first author introduces the frame of p-adic numbers defined with relations on the partially ordered set of open balls of the corresponding ultrametric of ${\mathbb{Q}}_p$ . As in the case of the reals, the first author explores the rings of p-adic continuous functions. This manuscript explores a modification of the construction of the frame of p-adic numbers $\mathcal{L}({\mathbb{Q}}_p)$ leading to the frame of the Cantor set. Set $|\mathbb{Z}{|}_p:=\{p^{-n+1}|n\in \mathbb{N}\}$ and let $\mathcal{L}({\mathbb{Z}}_p)$ be the frame generated by the elements $B_r(a)$ , where $a\in \mathbb{Z}$ and $r\in |\mathbb{Z}{|}_p$ , subject to the following relations:

(Q₁)

$B_r(a)\wedge B_s(b)=0$ whenever $|a-b{|}_p\ge r$ and $s\le r$ .

(Q₂)

$1=\bigvee \{B_r(a):a\in \mathbb{Z},r\in |\mathbb{Z}{|}_p\}$ .

(Q₃)

$B_r(a)=\bigvee \{B_s(b):|a-b{|}_p<r,s<r,b\in \mathbb{Z},s\in |\mathbb{Z}{|}_p\}$ .

Among several properties of this frame, we prove that it is spatial and isomorphic to the topology given by the ultrametric of the p-adic integers ${\mathbb{Z}}_p$ . Our main motivation for the study of this frame comes from a remarkable topological fact:

Theorem Hausdorff-Alexandroff

Every compact metric space is a continuous image of the Cantor space.

In this manuscript we provide a point-free version of this result (Theorem 4.17). There exist many proofs of the Hausdorff-Alexandroff Theorem in the literature, we take inspiration of a proof located in [2]; we implement a Hilbert-cube trick characterizing every compact metric frame as a closed quotient of the Hilbert cube frame and showing an injective frame morphism from the Hilbert cube frame to the frame of 2-adic integers.

In section 2 we present the necessary background used freely in our investigation.

Section 3 contains the presentation of the Cantor frame $\mathcal{L}({\mathbb{Z}}_p)$ . We give fundamental properties of this frame such as spatiality and perfectness. Also we observe that $\mathcal{L}({\mathbb{Z}}_p)$ is a complete uniform metrizable frame. At the end of this section, we give a direct characterization of $\mathcal{L}({\mathbb{Z}}_p)$ (Theorem 3.17).

In section 4 we give a point-free version of the Alexandroff-Hausdorff Theorem. We provide an injective morphism from the frame of $[0,1]$ into the frame of 2-adic integers via the dyadic rationals. We thus extend this morphism to an injective frame morphism from the Hilbert cube frame.

Section snippets

Preliminaries

A frame is a complete lattice L satisfying the distributive law $(\bigvee A)\wedge b=\bigvee \{a\wedge b|a\in A\}$ for any subset $A\subseteq L$ and any $b\in L$ . We denote its top element by 1 and its bottom element by 0. A frame is called spatial if it is isomorphic to $\mathrm{\Omega }(X)$ for some space X. A frame homomorphism is a map $h:L\to M$ satisfying $h(0)=0,h(1)=1,h(a\wedge b)=h(a)\wedge h(b),\text{ and}h\left(\underset{i\in J}{\bigvee }{a}_i\right)=\underset{i\in J}{\bigvee }h(a_i),$ where L and M are frames. Also, we say that h is dense if $h(x)=0$ implies $x=0$ .

The resulting category will be denoted by Frm. We have the natural functor

The frame of the Cantor set $\mathcal{L}({\mathbb{Z}}_p)$ and its properties

Recall that ${\mathbb{Q}}_p$ is the completion of $\mathbb{Q}$ with respect to its ultrametric. In [4], the frame of the p-adic numbers $\mathcal{L}({\mathbb{Q}}_p)$ is defined by generators and relations among open balls centered on rationals numbers. Besides, ${\mathbb{Z}}_p$ is the ring of integers of ${\mathbb{Q}}_p$ and it is a topological subspace which is homeomorphic to the Cantor set for every p, see [18]. Following the same construction in [4], we introduce the frame of the p-adic integers by replacing rationals with integers.

Definition 3.1

Let $\mathcal{L}({\mathbb{Z}}_p)$ be the frame generated

A point-free version of a theorem of Hausdorff-Alexandroff

In this section we will present a point-free counterpart of the following well known theorem due to Hausdorff and Alexandroff (see, e.g., [1] and [12]): $\mathit{\text{ Every compact metric space is a continuous image of the Cantor space}}$

As noted in [18], the map $\varphi :{\mathbb{Z}}_2\to [0,1]$ given by $\varphi \left(\sum _{i\ge 0}{b}_i{2}^i\right)=\sum _{i\ge 0}\frac{b_i}{2^{i+1}}$ is continuous and onto. We introduce some topological and set-theoretical properties of this map that motivates a frame-theoretical analogue.

Proposition 4.1

Let u be an integer coprime to 2, g a nonnegative integer where $0<u<2^{}$

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