A regular topological space $X$ has inductive dimension smaller or equal to n if and only if:
($n=-1$) $X=\emptyset$;
($n>-1$) The space $X$ has a base of opens $\mathscr{B}$ such that, for all $B\in\mathscr{B}$, $\delta B$ (the boundary of B) has inductive dimension smaller or equal to $n-1$.
Now.. if $X$ has inductive dimension smaller or equal to $n$, can we say something about the dimension of the boundary of a general open subset of $X$?