= Prevariety
In the classical convention over an <algebraically closed field> $k$, a prevariety is an <irreducible topological space> with a <sheaf of rings> which has a finite <open cover> by <affine varieties>, with compatible local identifications. A prevariety need not be separated. A classical <algebraic variety> is a separated prevariety: its <diagonal morphism> has closed image. Some authors allow reducible prevarieties and varieties; the separation criterion and product construction remain the same under that convention.
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