Let $M \subset \mathbb{R}^n$ be a compact connected manifold without boundary embedded in $\mathbb{R}^n$, then we can define the $\mathcal{C}^1$-topology of the functions $\mathcal{F}(M)= \{f: M\to \mathbb{R};$ $f$ is a smooth function$\}$, as the topology induced by the sets
$$B(f,\varepsilon) = \{ g: M \to \mathbb{R}; |f(x) -g(x)|<\varepsilon,\ \|\text{d}f_x - \text{d}g_x \|< \varepsilon,\ \forall \ x \in M\}. $$
where $$\|\text{d}f_x - \text{d}g_x \| = \sup_{v \in T_xM\setminus\{0\}} \left| (\text{d}f_x -\text{d}g_x)\left(\frac{v}{\|v\|}\right) \right|.$$ Let $A \subset M$ be a compact connected submanifold with boundary and same dimension of $M$.
My Doubt Is the $\mathcal{C}^1$- topology of $\mathcal{F}(A)$ equal the quotient topology induced by the function \begin{align*}\phi:\mathcal{F}(M) &\to \mathcal{F}(A) \\ f &\mapsto f\Big\vert_{A}\quad \quad? \end{align*}