$T: G \to X$ be a contraction mapping then $(I-T)(G)$ is an open subset of $X$.

Question

I was reading this proof from a book, in the last line of first image author considered the mapping $T_z: B(x_0;r)\to B(x_0;r)$ and then he's concluding the result from Banach fixed point theorem but for that we need a complete metric space but open ball is not complete so how can he use Banach fixed point theorem.