confused about the inverse function theorem in Rudin's textbook

Question

The confusion is highlighted with red rectangle.I don't understand the statement "it follows that $\varphi$ has at most one fixed point in $U$ ..."

It seems that "The Contraction Principle" cannot be applied here, since $\varphi(U)$ may not be a subset of U and U is open(U is not a complete metric space).

Answer 1

That is not an assertion on the existence of a fixed point, but just on its uniqueness. There cannot be two fixed points $x_1,x_2\in U$ because in that case $$\lvert \varphi(x_1)-\varphi(x_2)\rvert=\lvert x_1-x_2\rvert>\frac12\lvert x_1-x_2\rvert\ge \lvert \varphi(x_1)-\varphi(x_2)\rvert$$