Is the empty set considered a variety in affine and projective space? By variety, i mean a closed irreducible set in the Zariski topology.
On one hand it seems that the empty set satisfies the definition of e.g. an affine variety: it is an algebraic set and it is irreducible (to show that it is irreducible, we may argue that it can not be written as the union of two proper closed subsets simply because it does not have any proper subsets).
On the other hand, if the empty set is an affine variety, then the points of $\mathbb{A}^n$ are not minimal algebraic sets, since they contain the empty set.
Finally if we view the empty set as a projective variety in projective space, then we have an ambiguity since the empty set is both the zero set of the entire ring and the zero set of the maximal homogeneous ideal.
Any comments/flaws on my arguments above?
The empty set is a variety (it's the spectrum of the zero ring), but it's not an irreducible variety.
The usual definition of an irreducible variety does not do the right thing here, and one must use a slightly different one. An irreducible variety is a variety with exactly one irreducible component, and the empty variety has zero irreducible components.
This is the same kind of thing that causes $1$ to not be prime; the analogous definition of prime that does not work at $1$ is "a prime number is a positive integer which cannot be written as the product of two smaller positive integers." $1$ satisfies this definition but it still shouldn't be prime because that would make unique factorization false. See too simple to be simple for more details.