As the title suggests, the paper under review deals with extensions of topological spaces, where the extensions are constructed in the spirit of a one point extension, but the remainder can be larger and each point of it ``contributes'' to the extension of the original topology via a suitable ideal. Let \((X,\tau)\) be a topological space, let \((X\cup I,\sigma)\) be its extension (\(I\) is a nonempty set disjoint from \(X\), each nonempty element of \(\sigma\) hits \(X\) and the relative topology that \(X\) inherits from \((X\cup I,\sigma)\) coincides with \(\tau\)). Denote by \(\mathcal{C}(X)\) the closed subsets of \(X\). For each \(i\in I\), \(\{X\setminus W: W\in \sigma, i\in W\}\) is a closed base for an ideal \(\mathcal{B}_i(\sigma)\) on \(X\), and \(\mathcal{B}_i(\sigma)\)= \(\{E\subseteq X: i\notin cl_\sigma(E)\}\)= \(\{E\subseteq X: \exists W\in \sigma\) such that \(i\in W, W\cap E=\emptyset\}\). By the denseness of \(X\), the ideal is nontrivial. Thus, each extension \((X\cup I,\sigma)\) of \((X,\tau)\) gives rise to a family of nontrivial closed ideals \(\{\mathcal{B}_i(\sigma): i\in I\}\) on \(X\). The authors describe a natural way to associate an extension of \(X\) to every family of nontrivial closed ideals on \(X\) and study the properties of such (bornological) extensions. It is a beautiful and natural construction covering several known extensions (Alexandroff, Wallman, Stone-Čech, Dedekind-MacNeille, completion of a metric space) and, as the authors suggest in the concluding remarks, it should lead to topics outside general topology.