Let $X$ be a topological space and $\mathcal E$ and $\mathcal E'$ be two finite CW complex structures on $X$. Let $e\in \mathcal E$ and $e'\in \mathcal E'$ be two top dimensional cells such that $e\cap e'\neq \emptyset$. Assume further that $e'$ is not contained in $e$.
Question. Then is it true that $e'$ intersects the boundary of $e$?
(By the boundary of $e$ we mean $\bar e-e$.)
I think the answer to the above question is YES and here is my argument.
Denote the boundary of $e$ by $\partial e$. Suppose $e'\cap \partial e=\emptyset$. Note that $e'$ and $e$ are open in $X$ since they top dimensional cells. So we can partition $e'$ into two parts, one is $e\cap e'$ and the other is $(X-\bar e)\cap e'$. Both these are open in $X$. This is because $e\cap e'$ is an intersection of two open sets. Also, and $X-\bar e$ is open in $X$ because $\bar e$ is compact. Therefore $(X-\bar e)\cap e'$ is also open in $X$.
So we have written $e'$ as a union of two nonempty disjoint open sets which is a contradiction to the connectedness of $e'$.
Is this alright?