The following definition I found it in a text on Lie groups:
Let $M$ be a connected smooth manifold and $x_{0}\in M$. A path in $M$ starting at $x_{0}$ is a continuous curve $\gamma :[0,1]\longrightarrow M$ such that $\gamma (0)=x_{0}$. The space $P=P(x_{0},M)$ of all paths in $M$ starting at $x_{0}$, will be provided with the topology of uniform convergence.
My Question:
What is the topology of uniform convergence in this case?
I know it's this topology in the case where M is a metric space, but in this case where M is a manifold I can not say the same.