4. The Laplace-Beltrami Operator

Definition A Cantor set is a compact, completely disconnected set without isolated points Theorem Any Cantor set is homeomorphic to {0, 1}N. L. B, “On the structure of perfect sets of points”, Proc. Akad. Amsterdam, 12, (1910), 785-794. Hence without extra structure there is only one Cantor set. I.2)- Metrics Definition Let X be a set. A metric d on X is a map d: X × X 7 → R+ such that, for all x, y, z ∈ X (i) d(x, y) = 0 if and only if x = y, (ii) d(x, y) = d(y, x), (iii) d(x, y) ≤ d(x, z) + d(z, y). Definition A metric d on a set X is an ultrametric if it satisfies d(x, y) ≤ max{d(x, z), d(z, y)} for all family x, y, z of points of C. Given (C, d) a metric space, for > 0 let ∼ be the equivalenc