I am slowly encountering manifolds in my lectures, and am interested in the notion of open in relative topology with respect to manifolds:
Let $M\subseteq \mathbb R^{d}$ be an $n-$dimensional manifolds: it follows $\forall a \in M,\exists V\subseteq \mathbb R^{d}$ open in relative topology on $M$ and $\exists T \subseteq \mathbb R^{n}$ and a chart $\varphi: T \to V$
My main interest is the reason why we go to the lengths of including sets open in relative topology on $M$ rather than just open sets? What is the reason behind this?
I understand relative the topology of $M$, where $\gamma$ is the topology on $\mathbb R^{d}$, is described by $\tau_{M}:=\{M\cap B: B \in \gamma\}$