Abstract: In this survey lecture, I will mainly cover well-known results from a new angle. A Pervin space is simply a set equipped with a lattice of subsets. This notion suffices to define a natural notion of completion which appears to be the dual of the original lattice. One can then show that any lattice of subsets can be described by a set of inequations of the form u <= v, where u and v are elements of its dual space. Applications to formal languages and complexity classes will be given.