An ionad is yet another variant of the notion of topological space. The poset of open subsets and the category of sheaves on a topological space \(X\) only allow us to recapture the space’s sober reflection, not the actual points of \(X\). To remedy this the author defines an ionad to be a set \(X\) equipped with a finite limit preserving comonad \(I_X\) on the category \(\mathcal Set^X\) of \(X\)-indexed families of sets. We think of elements of \(X\) as the points of the ionad and the \(I_X\)-coalgebras as the sheaves. Many examples of ionads, including the ionad associated with a space, are provided by giving a basis; that is, a small category \(\mathcal B\) and a flat functor \(M:\mathcal B \rightarrow \mathcal Set^X\). If you are going to learn topology this way, with ionads, a little course on basic category theory might serve well. However, it will be worth it for the beauty of the subject’s development.