dg.differential geometry – Does the topological boundary of the manifold interior coincide with the manifold boundary?


Let $dinmathbb N$ and $M$ be a $d$-dimensional properly embedded (i.e. closed) $C^1$-submanifold of $mathbb R^d$ with boundary.

It can be shown that the manifold boundary $partial M$ and topological boundary $operatorname{Bd}M$ coincide. In the same way, the manifold interior $M^circ$ and topological interior $operatorname{Int}M$ coincide.

$M^circ$ is a $d$-dimensional embedded $C^1$-submanifold of $mathbb R^d$ without boundary and hence $partial M^circ=emptyset$.

However, can we show that $operatorname{Bd}M^circ=partial M$?