Are dimensions redundant? I am fairly new to this so apologies for informal terminology.
After I discovered what space filling curves are, I came to the conclusion that any point in any number of dimensions can be represented as a single number along a space filling curve, given that the curve covers enough space. I also imagine that it is possible to rewrite any operation on these points in terms of movements along the curve.
Is this correct to assume? 
 A: There are bijections between $\Bbb R$ and $\Bbb{R^N}$, and there are even isomorphism as Abelian groups between $\Bbb R$ and any vector space over $\Bbb R$ which has the same cardinality. At least assuming the axiom of choice.
But all those things are fairly destructive to the rest of the structure that we know and think of.
Equally, you could have said, that we don't need any countable structure, because it can be realized using $\Bbb N$. But there are still good reasons to think about $\Bbb Z$ and $\Bbb Q$ separately.
A: You could of course argue that there are function from $\mathbb R$ to $\mathbb R^n$ which are one-to-one and onto, so you could use them to "represent" points in $\mathbb R^n$ with a single coordinate.
However, although there are various different concepts of "dimension" they all have in common that it's not only about unique identification of points but also about being "natural" w.r.t. a certain structural/geometrical interpretation of the corresponding set/space.  The entry about "dimension" in the Princeton Companion to Mathematics might be a good starting point.
A: No. First, space filling curves are not one-to-one. And if you view the inverse as a multivalued function, it is highly discontinuous, rendering it useless for almost any purpose.
