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The space of continuous functions from a topological space to a complete one is always complete. The analyticity of a function in a closed condition in this topology (a uniform limit of holomorphic functions is holomorphic, and we can work locally if needed) so the space of holomorphic functions from the projective plane to itself is complete. This is precisely $\mathbb C(t)\cup\lbrace\infty\rbrace$, for $\infty$ the constant function equal to $\infty$.