 The positive topology on \(P\omega\) comprises finitely generated open sets. Let \(E\) denote the finite subsets of \(\omega\). \(X\subseteq P\omega\) is open iff \(X\) is a cone (with respect to \(\subseteq\)) generated by a subset of \(E\). Given the positive topology, a function \(f\colon P\omega\rightarrow P\omega\) turns out to be continuous (in the usual topological sense) iff \(f(x)=\cup\{f(e_n)\colon e_n\subseteq x\}\),  where \(e_n\in E\). This means that \(m\in f(x)\) iff \(\exists e_n\subseteq x.\, m\in f(e_n)\), which leads to a characterization of a continuous function  \(f\) by the pairs of natural numbers \((n,m)\).