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 that the Cantor set is homeomorphic to the p-adic integers for every prime number p. To give a point-free description of the Cantor set, we specify the frame by generators and relations. We use the fact that the open balls centered at integers generate the open subsets of 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 L(Z_p).

A topological characterization of it is given by Brouwer's Theorem: The Cantor set is the unique totally disconnected, compact metric space with no isolated points. We prove that the cantor frame is a spatial frame whose space of points is homeomorphic to Z_p. In particular, we show with point-free arguments that this frame is 0-dimensional, (completely) regular, compact, and metrizable. Moreover, we show that its Cantor-Bendixson derivative is zero. It follows that a frame L is isomorphic to L(Z_p) if and only if L is a 0-dimensional compact regular metrizable frame with zero Cantor-Bendixson derivative. 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. We prove the point-free analogue: if L is a compact metrizable frame, then there is an injective frame homomorphism from L into L(Z_p).