Space filling curve which is a closed map

Question

Does there exist a continuous surjective closed map $f : [0, 1] \to [0, 1]^d$? That is, does there exist a space-filling curve which is a closed map?

Answer 1

Since $[0,1]$ is compact, each its closed subset $C$ is compact, so $f(C)$ is compact for any continuous map $f$ from $[0,1]$ to a topological space $X$. If the space $X$ is Hausdorff, then $f(C)$ is closed and thus $f$ is closed.

Answer 2

The Cantor function maps the Cantor set $C\simeq \{0,1\}^\omega$ onto $[0,1]$. So its $d$-fold product, maps $C^d$ onto $[0,1]^d$ and $C^d \simeq C$ so we have a continuous map $C \to [0,1]^d$ which is onto. Now extend to $[0,1]$ by the generalised Tietze theorem or by linear interpolation (as in the original Cantor map). So we have a continuous "dimension-raising" map $[0,1] \to [0,1]^d$. It's automatically closed too, as the domain is compact and the co-domain is Hausdorff.