The small inductive dimension $\operatorname{ind}X\in\{-1,0,\ldots\,\infty\}$ of a topological space $X$ is defined inductively, as follows.
- If $X=\varnothing$, then $\operatorname{ind}X=-1$.
- If for every $x\in X$ and its open neighborhood $U$ there is an open neighborhood $V$ satisfying $\operatorname{cl}V\subseteq U$ and $\operatorname{ind}\partial V\le n-1$, then $\operatorname{ind}X\le n$.
- If $\operatorname{ind}X\le n$ and $\operatorname{ind}X\not\le n-1$, then $\operatorname{ind}X=n$. If $\operatorname{ind}X\not\le n$ for any finite $n$, then $\operatorname{ind}X=\infty$.
So, a topological space is zero-dimensional with respect to $\operatorname{ind}$ if and only it has a base consisting of clopen sets. A topological space $X$ is said to satisfy the finite sum theorem for the small inductive dimension $\operatorname{ind}$ (here FSTi for short), if for every two closed sets $A,B\subseteq X$ we have $$\operatorname{ind}(A\cup B)\le\max\{\operatorname{ind}A,\operatorname{ind}B\}.$$ The dimension $\operatorname{ind}$ has the property that $\operatorname{ind}Y\le\operatorname{ind}X$ for any $Y\subseteq X$ and any topological space $X$, so the inequality in the definition above can be replaced by an equality. In an exercise of Charalambous's Dimension theory, the author says that if topological spaces $X,Y$ both satisfy FSTi, we have $$\operatorname{ind}(X\times Y)\le\operatorname{ind}X+\operatorname{ind}Y.$$ This makes me wonder what kinds of topological spaces satisfy FSTi. By searching I have only found out that some classes of spaces satisfy some similar properties that do not seem (for me) enough to imply FSTi. For example, $$\operatorname{ind}(A\cup B)\le\operatorname{ind}A+\operatorname{ind}B+1$$ if $A,B$ are subsets and $X$ is completely normal (Hausdorffness is not assumed here), and $$\operatorname{ind}(A\cup B)\le\max\{\operatorname{ind}A,\operatorname{ind}B\}+1$$ if $A,B$ are closed and $X$ is an arbitrary space. The author also gives examples of a metrizable space and a compact Hausdorff space not satisfying FSTi. Every space with small inductive dimension zero obviously satisfy FSTi. But the product inequality for them only says that a (finite) product of zero-dimensional spaces is zero-dimensional, which follows from the fact that $U\times V$ is clopen if $U,V$ are clopen and which is valid also for infinite products. By the way, the product inequality for $\operatorname{ind}$ holds for separable metrizable $X,Y$, since the product inequality for the covering dimension $\dim$ holds for metrizable $X,Y$, and $\dim=\operatorname{ind}$ in any separable metrizable space.
My questions in summary: which classes of topological spaces are known to satisfy FSTi? Conversely, which properties do spaces with FSTi have to satisfy, besides the product inequality?