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I have a concern about a definition of a quasi-affine variety. I had a professor who defined a quasi-affine variety to be an intersection of an open set and a closed set in some affine space $\mathbb{A}^n$, and an affine variety was a quasi-affine variety isomorphic to a closed set.

However, I think the more common definition is that given in Hartshorne, where a quasi-affine variety is an open subset of an affine variety, which is an irreducible closed subset of $\mathbb{A}^n$.

Are these definitions equivalent? Or have I been learning things slightly differently?

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Well, the difference between the two definitions you've given is that the former could be reducible whereas the latter will not be. In general there's some disagreement between authors as to whether "irreducible" is part of the definition of "variety." Other than that, they're the same.

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