Doubt in the definition of closed subvarieties

Question

I'm trying to understand this definition in Fulton's algebraic curves:

In order to be $Y$ a variety, $\overline Y$ has to be an irreducible algebraic set of $\mathbb P^{n_1}\times\ldots\times\mathbb A^m$. The author shows $\overline Y$ is irreducible, but not algebraic set. How can we prove $\overline Y$ is an algebraic set? is this really necessary?

I need help

Thanks in advance

Answer 1

The closed sets in the Zariski topology are exactly the algebraic sets. Thus, the closure $\overline{A}$ of any subset $A$ of $X$ is always an algebraic set.