TITLE: Surjection that increases dimensions
QUESTION [5 upvotes]: This question is somewhat inspired by a question on MathOverflow, but it is not necessary to read that question to understand what I am about to ask. 
It is well known that one can establish a surjection between sets of different Hausdorff dimensions: in the regime of just set theory the cardinality of the unit interval and the unit square are the same, and in fact we get a bijection. If you add a bit of topology, one can in addition request that this surjection be given by a continuous map, but the map cannot be a bijection, else it'd be a homeomorphism. 
What if, instead of continuity, we require a different condition? 
Question Fix $N$ a positive integer. Let $B$ be the open unit ball in $\mathbb{R}^N$. Can we find an embedded smooth (or $C^1$) hypersurface $A\subset \mathbb{R}^N$ and a surjection $\phi:A\to B$ such that the vector $a - \phi(a)$ is orthogonal to $A$? Can it be made continuous? Can it be made a bijection?

REPLY [2 votes]: If $A$ is a hypersurface (co-dimension one and smooth) what you're describing is the graph of a function on $A$ -- well, locally that's what it is.  But the problem boils-down to a local problem.  You're asking for functions $f : D^{n-1} \to \mathbb R$ whose graph is an open subset of $\mathbb R^n$.  This isn't possible, even if $f$ is discontinuous.