closure of a prime ideal (zariski topology)

Let \(A\) be a commutative ring, \(\mathfrak{p} \in \mathrm{Spec}(A)\) a prime ideal and \(\mathrm{Spec}(A)\) equipped with the Zariski topology. Then

\begin{align*} \overline{ \{\mathfrak{p}\}} = V(\mathfrak{p}) \end{align*}

By characterization of the closure as smallest closed set we get

\begin{align*} \overline{ \{\mathfrak{p}\}} = \bigcap_{T \subseteq \mathfrak{p}} V(T) \end{align*}

Therefore we get by applying vanishing set of a union

\begin{align*} \bigcap_{T \subseteq \mathfrak{p}}(V(T)) =& V(\bigcup_{T \subseteq \mathfrak{p}} T) \\ =& V(\mathfrak{p}) \end{align*}