mapping low dimensional space to high dimensional space injectively?

Question

Is it possible to find an injective mapping that transforms low dimensional space like $\mathbb{R}$ to a subset of $\mathbb{R}^2$ with interior points? For instance, can we find a mapping from $\mathbb{R}$ to a unit disk in $\mathbb{R}^2$ ($x^2+y^2\leq 1$)?

Answer 1

I won't answer your second question (to allow you some time to think out what kind of function would be an injection), but one obvious map that does this is the map that takes the real number $x$ and associates it with the point $(x,x)$ $\in$ $\mathbb R^2$. You can easily verify that this function is indeed injective.