Hartshorne chapter II questions about structure sheaf.

Hartshorne chapter II questions about structure sheaf.

I am reading Hartshorne algebraic geometry. On page 71, the paragraph
before Proposition 2.2, it is said that "if $V(\mathfrak{a})$ is a closed
set" and $D(f) \cap V(\mathfrak{a}) = \emptyset$. But it seems that
$V(\mathfrak{a})$ is always closed. In this paragraph, it is proved that
for each $\mathfrak{p} \not\in V(\mathfrak{a})$, there is an $f$ such that
$\mathfrak{p} \in D(f)$. Could we conclude that $D(f) \cap V(\mathfrak{a})
= \emptyset$? Thank you very much.