Homeomorphism Compact Subsets Are there compact subsets $A,B \subset \mathbb{R^2}$ with $A$ not homeomorphic to $B$ but $A \times [0,1]$ homeomorphic to $B \times [0,1]$?

This is an addendum to the nice example below.
It was proven by K. Borsuk (Theorem 16 in the paper "On the decomposition of a locally connected compactum into Cartesian product of a curve and a manifold", Fund. Math. 40, (1953) 140–159.) that if X, Y are locally connected compacta of dimension at most 1, and their products with interval are homeomorphic then X is homeomorphic to Y. In fact, instead of an interval, one can use any topological manifold with or without boundary.  
