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  "body": "\n## Definition\n\nLet $k$ be an [[algebraically closed field|0PF6TM8TWT]].\nAn *algebraic variety* over $k$ (in the classical sense) is one of the following, with its Zariski topology:\n\n1. an [[affine variety|JERVQT4C12]], that is, an irreducible algebraic set in $k^n$;\n2. a quasi-affine variety, that is, a nonempty open subset of an affine variety;\n3. a [[projective variety|A4H4CZ2V35]], that is, an irreducible projective algebraic set in $\\mathbb{P}^n$;\n4. a [[quasi-projective variety|EQSHJPE7H8]], that is, a nonempty open subset of a projective variety.\n\n## Remarks\n\nEach of the four kinds is a quasi-projective variety, so the definition says that a variety is a quasi-projective variety.\nEvery variety is [[irreducible|SY2MWC434Z]] and its topology is Noetherian.\n\nExample: the affine line $k$, the projective plane $\\mathbb{P}^2$, and the punctured plane $k^2 \\setminus \\set{(0, 0)}$ are varieties.\n\nNon-example: $V(xy) \\subseteq k^2$ is not a variety in this sense, since it is reducible.\n\nOther conventions exist.\nSome authors allow reducible varieties.\nIn the language of schemes, a variety over any field $k$ is often an integral separated scheme of finite type over $k$.\nThis includes abstract varieties that are not quasi-projective.\n",
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