In analogy to a property used by R. Freese and J. B. Nation [Pac. J. Math. 75, 93–106 (1978; Zbl 0382.06005)] to characterize projective lattices the authors introduce for a Tychonoff space \(X\) the FN property which means that \(X\) has a base \(\mathcal{B}\) such that for every \(V\in\mathcal{B}\) there are two finite sets \(u(V)\subseteq\{U\in\mathcal{B}:V\subseteq U\}\) and \(l(V)\subseteq\{U\in\mathcal{B}:V\supseteq U\}\) such that \(u(V)\cap l(W)\neq \emptyset\) whenever \(V\subseteq W\). \(X\) has the FNS property if \(X\) has a base \(\mathcal{B}\) with the FNS property, i.e. there exists a map \(s:\mathcal{B}\rightarrow [\mathcal{B}]^{<\omega}\) such that if \(U,V\in\mathcal{B}\) are disjoint, then there are disjoint sets \(W_U,W_V\in s(U)\cap s(V)\) such that \(U\subseteq W_U\) and \(V\subseteq W_V\). The FNS property implies the FNS\(^*\) property, i.e. there is a \(\pi\)-network with the FNS property.
The main results are: The product of spaces with the FN property (the FNS property, the FNS\(^*\) property) has, respectively, the FN property (the FNS property, the FNS\(^*\) property). Every space with the FNS\(^*\) property satisfies the countable chain condition. Every metrizable space has the FN property. Moreover, the authors give a new proof for Heindorf and Shapiro’s result that for a 0-dimensional openly generated space \(X\) the family of clopen sets \(\text{Clop}(X)\) has the FNS property.
For Part II, see [the authors, Topology Appl. 304, Article ID 107784, 11 p. (2021; Zbl 1479.54052)].