Let $mathbb{F}$ be an algebraically closed field. Consider the following definition of the dimension of a (quasi)projective $mathbb{F}$-variety, given in Harris *Algebraic Geometry: A First Course*:

It seems nonstandard to take this as the *definition* of a variety, so I will think of this as a theorem that should be proven from a more conventional definition of dimension, e.g. the Krull dimension.

My question is this: Does Definition 11.2 generalize to structures other than a variety over an algebraically closed field? For example, what if $mathbb{F}=mathbb{R}$? Or $X$ is a manifold embedded in projective space? In these cases, does the dimension of $X$ still agree with the smallest codimension of a disjoint linear subspace?

Note that Definition 11.2 holds when the “irreducible” assumption is dropped, so this small generalization does hold.