The union of two affine varieties can be expressed as $ \mathbb{V}(\{F_i\}_{i\in I}) \cup \mathbb{V}(\{F_j\}_{i\in J}) = \mathbb{V}(\{F_iF_j\}_{(i,j) \in I\times J}). $ We want to generalize this to an infinite union. Let $\{A_i\}_{i \in I}$ be an (infinite) partition of the indexing set $A$; then the left-hand side becomes $ \bigcup_{i \in I} \mathbb{V}(\{F_\alpha\}_{\alpha \in A_i}), $ while the defining 'polynomials' on the right hand side must be products of infinitely many polynomials. However, any polynomial must be a finite sum of monomials, so an infinite union of affine varieties is never an affine variety.
Is this correct? If not, are there examples of infinite unions that are varieties, and infinite unions that are not?