Is there Domain invariance for manifolds with boundary in some sense?

Question

It is well known that for manifolds without boundary, there exist a domain invariance theorem in the following form.

Theorem. A subspace in an $n$-dimensional manifold without boundary is open if and only if it is an $n$-dimensional manifold without boundary.

I am curious on the case of manifolds with boundary. That is

Suppose $M$ is an $n$-dimensional manifold with boundary, then whether $A\subset M$ is an $n$-dimensional manifold without boundary if and only if it is an open subspace contained in the interior of $M$?

Answer 1

We only need to proof that if $A$ is an $n$-manifold without boundary, then it is open in the $M$.

In fact, the answer of this question is very simple. The idea comes from Anton Petrunin, see https://mathoverflow.net/questions/191422.

Let's construct a new manifold without boundary by doubling $M$, denoted by $\widetilde M$. By the domain invariance for manifolds, $A$ is an open subspace in $\widetilde M$. Hence it is open in $M$.

Answer 2

This is false. For a 1-dimensional counterexample, take $M=[0,1]$ and $A=[1/4,3/4]$.