# 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 $$operatorname{Bd}M^circ=partial M$$?