Collections of higher-dimensional hereditarily indecomposable continua, with incomparable Fréchet types

We construct a family {Ys: s∈S} of cardinality 2ℵ0 of hereditarily indecomposable continua which are: (a) n-dimensional Cantor manifolds, for any given natural number n, or (b) hereditarily strongly infinite-dimensional Cantor manifolds, or else (c) countable-dimensional continua of every given transfinite inductive dimension, small or large, such that if h :Ys→Ys′ is an embedding then s=s′ and h is the identity.