Appendix C - Cantor spaces

Summary

It is well–known that a compact metrizable space X is homeomorphic to {0, 1}^ℕ if and only if X is nonempty, perfect and totally disconnected (hence, zero–dimensional). The classical Cantor ternary set in ℝ is one such, thus the name Cantor spaces. There are many other classical examples. A useful one is the product space k^ℕ, where k is a finite space endowed with the discrete topology and with card(k) > 1. It will be necessary that Cantor spaces be investigated not only as topological spaces but also as metric spaces with suitably assigned metrics.

The development presented in this appendix is based on E. Akin [2], R. Dougherty, R. D. Mauldin and A. Yingst [47], and O. Zindulka [162, 161]. There are two goals. The first is to present specific metrics on Cantor spaces which are used in the computations of Hausdorff measure and Hausdorff dimension in Chapter 5. The second is to discuss homeomorphic measures on Cantor spaces. The lack of an analogue of the Oxtoby–Ulam theorem for Cantor spaces motivates this goal.

Topologically characterizing homeomorphic, continuous, complete, finite Borel measures on Cantor spaces is a very complex task which has not been achieved yet. Simple topological invariants do not seem to characterize the homeomorphism classes of such measures. By introducing a linearly ordered topology consistent with the given topology of a Cantor space, which is always possible, a linear topological invariant has been discovered by Akin in [2].