Note that \(\mathbb{B}:=\{p\} \cup \{\{p,q\}: q\in X,q\neq p \}\) is a basis. We check that the topology \(\Omega_{B}\) generated by \(\mathbb{B}\) is the VIP topology on \(X\). Let \(U\) be a subset of \(X\) containing \(p\). If \(x \in U\) then choose \(B=\{p\}\)if \(x=p\), and \(B=\{p,x\}\) otherwise. Note further that if \(p\notin U\)then there is no \(B \in \mathbb{B}\) such that \(B \subseteq U\). This shows that \(\Omega_{\mathbb{B}}\) is precisely the VIP topology on \(X\).