This terminology question has always tripped me up: If $M$ is a topological manifold with boundary, the (nontrivial!) result that $\partial M$ and $\operatorname{Int}{M}$ are disjoint is called "Invariance of the Boundary" (at least in John Lee's Introduction to Smooth Manifolds). Why this name? When I see the word "invariance" I think of a property unchanged by a mapping. For example, "invariance of the boundary", to me, would mean something more like "If $M$ and $N$ are manifolds with boundary and $f : M \to N$ is a homeomorphism, then $\partial M = \partial N$ and $\operatorname{Int}{M} = \operatorname{Int}{N}$." But we don't need the fact that the boundary and interior of a manifold are disjoint to prove this.
Is there another reason why we call the disjointness result "invariance of the boundary"?