Let $M$ be a $d$-dimensional manifold in $\mathbb{R}^p$ and let $r>0, x \in \mathbb{R}^p$. Then $N:=rM+x$ is also a $d$-dimensional manifold in $\mathbb{R^p}$.
I want to show that for $B \subseteq M \mathcal{}\\B \in \mathcal{B}(M) \Leftrightarrow (rB+x) \in \mathcal{B}(N)$
where $\mathcal{B}$ is the Borel $\sigma$ algebra.
How can I show this? I thought about using the fast that $T(B)=rB+x$ is a homeomorphism somehow but I don't know if that works
Any homeomorphism $T$ between topological spaces $X$ and $Y$ preserves Borel sets in the sense $B$ is Borel in $X$ iff $T(X)$ is Borel in $Y$. This is easy to see from the definition of Borel sigma algebra as the one generated by open sets. In this case $Ty=ry+x$ defines a homeomorphism.