Uniform Approximation of Topological Spaces

We sharpen the notion of a quasi-uniform space to spaces which carry with them functional means of approximating points, opens and compacts. Assuming nothing but sobriety, the requirement of uniform approximation ensures that such spaces are compact ordered (in the sense of Nachbin). We study uniformly approximated spaces with the means of topology, uniform topology, order theory and locale theory. In each case it turns out that one can give a succinct and meaningful characterization. This leads us to believe that uniform approximation is indeed a concept of central importance.