Question: Let $M$ be a smooth manifold and $S_1,S_2\subseteq M$ two locally-closed submanifolds (i.e. they are open in their closure). If $$S_1\subseteq\overline{S}_2-S_2,$$ is it true that $$\dim S_1<\dim S_2?$$
I believe it is true since it seems that it is used implicitly in this paper (the sentence between Definitions 1.5 and 1.6). But I couldn't find a way to prove it rigourously.