# A Tietze-like Theorem

Suppose $X$ is a normal (Hausdorff) zero dimensional topological space and $C\subset X$ is a closed set. Let $f:C\rightarrow \mathbb N$ be continuous. By the Tietze extension theorem, there exists a continuous $g:X\rightarrow \mathbb R$ such that $f\subset g$. I want to know if we can guarantee that there exists a continuous $g:X\rightarrow \mathbb N$ such that $f\subset g$ (even if f is not bounded).

I know that in general it's not true since that would imply that every normal topological space that has at least two points is not connected, therefore I added the zero dimensional hypothesis.

Edit: If it's not true, can I fix it by adding an extra hypothesis?

According to this paper, theorem 6, it is true for so called ultranormal spaces: spaces where any two disjoint closed sets are separated by a clopen set. Then we can even take any complete metric space as the codomain, including $\mathbb{N}$. This condition is clearly necessary (for any two disjoint closed non-empty sets extend the map that sends one closed set to $0$ and the other to $1$, which is a continuous map into $\mathbb{N}$ and take the inverse image of $\{0\}$). Not all zero-dimensional normal spaces are ultranormal (Roy has constructed a classical counterexample, to which the paper also refers.)