Certain 2-stable embeddings

The Chogoshvili Claim states that for each k-dimensional compactum X in ℝⁿ, there exists an (n - k)-plane P in ℝⁿ such that X is not removable from P. This means that for some ε > 0, every map f : X → ℝⁿ with ∥x - f(x)∥ < ε for all x ∈ X, has the property that f(X) ∩ P ≠ Ø. A counterexample to this claim has recently been constructed by A. Dranishnikov. Our results show, among other things, that each 2-dimensional LC¹ compactum, and hence each 2-dimensional disk, obeys the claim. To help indicate the sharpness of the preceding, we also provide a local path-connectification of Dranishnikov's example.