A minimal structure \(m\) on a set \(X\) is a subset of the power set of \(X\) such that \(\varnothing,X\in m\). Compactness and continuity of topological spaces, as well as a number of other notions such as Hausdorff and regular, extend in the obvious way to minimal structures by replacing the topology by any minimal structure. This paper looks at criteria for some of these properties as well as interactions between them.