Mappings between Euclidean spaces that are one-to-one over the image of a dense subset

Abstract:

Examples are constructed that include, for $n > m \geqslant 2$, proper surjective maps $f:{{\mathbf {R}}^n} \to {{\mathbf {R}}^m}$ that are one to one over the image of a dense subset $D \subset {{\mathbf {R}}^n}$; i.e., the restriction of $f$ yields a homeomorphism from ${f^{ - 1}}f(D)$ onto $f(D)$. A routine Baire Category argument establishes that, for any $\sigma$-compact subset $F \subset {{\mathbf {R}}^n}$ which $f$ maps onto ${{\mathbf {R}}^m}$, ${\operatorname {Int}}(F) \ne \theta$ and, hence, $\dim F = n$.