Monotone basic embeddings of hereditarily indecomposable continua

Let {Φ_i}_i=1^κ be monotone maps on a hereditarily indecomposable continuum X. It is proved that the following are equivalent: (i) The product map Φ = (Φ₁, Φ₂, ..., Φ_κ) is light, (ii) Φ is an embedding. (iii) Each f in C(X, R) is representable as f = Σ_i=1^κg_ioΦ_i with g_i ∈ C(Φ_i(X), R). This is applied to prove the following result which is related to the Chogoshvili conjecture: Let n ≥ 2 and let X be an n-dimensional hereditarily indecomposable continuum. X can be embedded in a separable Hilbert space H such that: (i) The restriction to X of the continuous linear functionals of H forms a dense subset of C(X, R). (ii) There exists an orthonormal basis B for H such that the restriction to X of each 2-dimensional B-coordinate projection of H factors through some 1-dimensional space and as a result has no stable values in R². In particular the n-dimensional B-coordinate projections have no stable values on X.