If $T$ is a continuous bijection on $\mathbb R^d$, and $M$ is a $C^1$-submanifold with boundary, is $T(\partial M)=\partial T(M)$?

Question

Let $d\in\mathbb N$ and $T:\mathbb R^d\to\mathbb R^d$ be bijective and continuous. I'm not sure how to approach this, but if $M\subseteq\mathbb R^d$, does $T$ map the topological boundary/interior of $M$ onto the topological boundary/interior of $N:=T(M)$? And if $M$ is an embedded $C^1$-submanifold of $\mathbb R^d$ with boundary, does $T$ map the manifold boundary/interior to the manifold boundary/interior of $N$? Or do we need further assumptions?