In Algebraic Geometry one has a very useful formula for the dimension of fibers. Specifically I am thinking about a statement of the following form:

Let $C$ be a curve over $\mathbb{C}$, and let $S$ be an irreducible constructible subset $S \subset C^n$. Here we take the Zariski topology on $C^n$ and constructible means relatively closed $S \subset_{cl} U \subset_{op} C^n$, irreducible means there are no constructibles $S_1, S_2 \subset_{cl} S$ such that $S = S_1 \cup S_2$ and $S_1 \not\subset S_2$, $S_2 \not\subset S_1$. Let $\pi: C^n \rightarrow C^m$ be a projection. Then $$\dim S = \dim \pi(S) + \min_{a \in \pi(S)} \dim(\pi^{-1}(a) \cap S).$$

I wonder, is there a similar statement for topological dimensions?

In particular, consider a compact Hausdorff space $X$ (or maybe seperable metric or similar) and a projection $\pi : X^n \rightarrow X^m$. For which subsets $S \subset X^n$ do we have $$\dim S = \dim \pi(S) + \min_{a \in \pi(S)} \dim(\pi^{-1}(a) \cap S)?$$

I don't know a lot about topological dimension theory, but some very basic first examples with the unit cube $I^n$ seem to work out for constructible subsets, i.e. locally compact subsets here.

Also note that if $Y \subset X^n$ has $\dim Y = 0$, then $\dim \pi(Y) = 0$ because $\pi$ is continuous open-and-closed.

Any ideas, help, or pointers to literature greatly welcome!