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 Z p : = { x ∈ Q p : | x | p ≤ 1 } for every prime number p . To give a point-free description of the Cantor set, we specify the frame of Z p by generators and relations. We use the fact that the open balls centered at integers generate the open subsets of 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 L ( 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 L ( Z p ) is a spatial frame whose space of points is homeomorphic to Z p . In particular, we show with point-free arguments that L ( Z p ) is 0-dimensional, (completely) regular, compact, and metrizable (it admits a countably generated uniformity). Moreover, we show that the frame L ( Z p ) satisfies c b d L ( Z p ) ( 0 ) = 0 , where c b d L ( Z p ) : L ( Z p ) → L ( Z p ) , defined by c b d L ( Z p ) ( a ) = ⋀ { x ∈ L ( Z p ) | x ≤ a and ( x → a ) = a } is the Cantor-Bendixson derivative (see, e.g. [19] ). 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 c b 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 L ( Z 2 ) .