= Cross-ratio preserving maps are projective
An <injective> self-map of the complex <projective line> preserving all <cross-ratios> is a <projective transformation>; neither continuity nor prior surjectivity is needed. Choose the unique projectivity agreeing with the map on three distinct <projective points> and compose with its inverse. The resulting map fixes those three points. Normalize them to $\infty,0,1$. Preservation of the <cross-ratio> $(\infty\,0:1\,z)=z$ now fixes every other point as well, proving that the normalized map is the <identity map>. Undoing the composition proves the assertion.
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