Intuition behind the Sum Theorem in dimension theory

Question

In the Menger-Urysohn dimension, for any two subspaces $A, B$ of $X$, there is $$ \operatorname{dim}(A\cup B)\leqslant \operatorname{dim}(A)+\operatorname{dim}(B)+1 $$ However, if $A, B$ are closed subspaces with $\operatorname{dim}(A)\leqslant n$ and $\operatorname{dim}(B)\leqslant n$, then $$ \operatorname{dim}(A\cup B)\leqslant n $$ This is part of the Sum Theory in dimension theory. I wonder what is the intuition behind this fact. Why do closed sets not increase dimension?