Consider a continues curve $\Gamma^*:r(t), t\in [0,1]$, in $\mathbb{R}^d$. Suppose $\Gamma^*$ is self-intersected. Does $\Gamma^*$ can be uniformly approximated by a sequence of non-self-intersect curves $\Gamma_n:r_n(t), t\in [0,1], n=1,2,....$?
If $d=2$, the answer is no. An example: the curve '4'.
How about $d \ge 3$? I think the answer is yes. But I am not sure, especially when $\Gamma^*$ has infinite self-intersection points and it is space filling. -- It is yes by using the polygonal curves.
More general cases: Consider a continues function $f^*: [0,1]^m \to \mathbb{R}^d, d>m$. Does $f^*$ can be uniformly approximated by a sequence of non-self-intersect functions $f_n:[0,1]^m \to \mathbb{R}^d, n=1,2,....$?