The wikipedia article for semi-recursive sets (formally titled "recursively enumerable sets") claims:
A set S of natural numbers is called recursively enumerable if there is a partial recursive function whose domain is exactly S, meaning that the function is defined if and only if its input is a member of S.
The set S is the range of a total recursive function or empty. If S is infinite, the function can be chosen to be injective.
Why in the infinite case can we demand injectivity for the function?