Starting from the topological point of view a certain wide class of T-spaces is introduced having a very strong extension property for continuous functions with values in these spaces. It is then shown that all such spaces are complete lattices whose lattice structure determines the topology -these are the continuous lattices -and every such lattice has the extension property. With this foundation the lattices are studied in detail with respect to projections, subspaces, embeddings, and constructions such as products, sums, function spaces, and inverse limits. The main result of the paper is a proof that every topological space can be embedded in a continuous lattice which is homeomorphic (and isomorphic) to its own function space. The function algebra of such spaces provides mathematical models for the Church Curry λ-calculus.