In the book by Frankel The geometry of physics on page 25 there is a definition of coordinate curve as follows: $x^i$(t)=constant for $i\neq \alpha$, and $x^\alpha(t)=t$. I do not understand why both, the argument and value of $x^i$, $t$, lie on $R^1$. I.e. one of them (either the argument of $x^i$ or the value) should lie in an open set $U$ of the manifold $M^n$! Can someone describe the precise what we are "really" dealing with? If you don't have the book, just type
"coordinate curve through a point,the curve being parameterized by"
into google,and follow the first link.