I am not a mathematician (so please be kind :) but I need to understand this problem in relation to my work in a different field.
Let $\mathcal X,\mathcal Y,\mathcal Z$ be topological spaces with sets $X,Y,Z$. Let $\mathcal Y$ and $\mathcal Z$ be "nice", compact - no pathologies, no isolated points, etc., however not necessarily metric spaces. Initially, suppose $\mathcal Y$ and $\mathcal Z$ are homeomorphic and have dim n. My question is, under this condition (and also when $\mathcal Y$ and $\mathcal Z$ are not of the same dimension) what can be said about the topological dimension of the set $(A\cup B) \subset X$ and of the space $\mathcal X$ when they are defined as follows:
Define $X$ as $X = Y \cup Z$, where $Y \cap Z = \emptyset$.
Let $\mathcal X$ be a connected space such that, although disjoint, $Y$ and $Z$ each contain limit points of the other. Let $A \subseteq Y$ and $B \subseteq Z$ be the set of all limit points of $Z \in Y$ and the set of all limit points of $Y \in Z$, respectively and let $A \neq \emptyset$ and $B \neq \emptyset$.
I think the following are true:
$Y$ and $Z$ are neither open nor closed in $X$.
$Y\setminus A$ and $Z\setminus B$ are open in $X$
$(A\cup B) \subset X$ is a boundary between $(Y\setminus A) \subset X$ and $(Z\setminus B) \subset X$.
$A\cup B$ is closed in $X$.
$A\cup B$ has no interior.
$\overline A = A\cup B$ and $\overline B = A\cup B$.
$A$ is dense in $A\cup B$ and $B$ is dense in $A\cup B$.
$A$ and $B$ are mutually dense. Every point of $A$ is a limit point $B$ and every point of $B$ is a limit point of $A$.
My attempt at proving these things is here: https://sidereal.com/ndproof.pdf. It appears to me that $A \cup B$ is a "fine partition" as defined in Encyclopedia of Mathematics here: https://encyclopediaofmath.org/wiki/Partition.
Intuitively it seems to me that $A\cup B$ should have a defined dimension at least in the case that dim $\mathcal Y$ and dim $\mathcal Z$ are equal, but since it is closed with no interior I cannot see how to approach the question. Given that, I am then unsure how to deal with the dimension of $\mathcal X$.
Your help would be greatly appreciated.
Added note: The goal here is to understand the nature of the space $\mathcal X$ given the structure of $A \cup B$. If initial restrictions on the properties of the spaces $\mathcal Y$ and $\mathcal Z$ must necessarily be dropped in this case, then knowing that would be a step forward. The intent would be that those properties then would be recovered when $\mathcal Y$ and $\mathcal Z$ are then considered as subspaces of $\mathcal X$ with the subspace topology.