# A confusion regarding the definition of a quasi-affine variety.

A quasi-affine variety is an open subset of an affine variety.

Open under Zariski topology? How does this make sense?

• Meaning "relatively open". So a quasi-affine is the intersection of an affine variety with an open subset of the ambient affine space. The usual example is $\mathbf{A}^2 - \{(0, 0)\}$. – Hoot Dec 31 '14 at 17:08
• @Hoot- Would $x+y=0\cap [(0,1)\times(2,3)]$ be another example? – algebraically_speaking Dec 31 '14 at 17:10
• @algebraically_speaking: No that set would not be quasi-affine: you have to intersect with a Zariski-open set, not a Euclidean-open set. The set $(0,1)\times (2,3)$ is not Zariski-open because its complement isn't the solution set of a polynomial. Hoot: He's referring to a piece of the line $y=-x$, intersected with an open rectangle. – pre-kidney Dec 31 '14 at 17:50

Yes, open under the Zariski topology. Here is how it makes sense: $$An \ \text{affine} \ variety \ is \ the \ solution \ set \ of \ polynomials.$$
$$A \ \text{quasi-affine} \ variety \ is \ a \ solution \ set \ of \ polynomials \ minus \ another \ solution \ set.$$