I'm reading the article "Local and simultaneous structural stability of certain diffeomorphisms - Marco Antônio Teixeira", and on the first page says "Denote $G^r$ the space of germs of involution at $0$ with the $\mathcal{C}^r$ topology."
I know how the $\mathcal{C}^r$-Whitney topology works, but I don't have the faintest idea what is the $\mathcal{C}^r$-germ topology. So I tried to search online about the definition of the $\mathcal{C}^r$ topology in the germ space, but I was not able to find anything. Can anyone please explain how this topology is defined or indicate a reference to me, so I can learn about it?
I have sent a message to Marco Antonio (the one who wrote the paper) and he answered me. The $\mathcal{C}^r$-germ topology is the topology regenerated by the basis
$$\mathscr{B}^r(\varphi,\varepsilon) =\left\{\psi \in G^r;\ \exists \mbox{ a neighborhood}\ U^{\psi}\ \mbox{of }0,\ \mbox{such that }\sup_{x\in U^\psi \\0\leq i \leq r}\|\text{d}^i\varphi_x - \text{d}^i\psi_x \|<\varepsilon\right\}, $$
where $\varphi \in G^r$ and $\varepsilon >0.$