The intuitive picture I have of giving a set a topology, is that of giving it a shape in the sense of connecting the points and determining what points lie next to each other (continuity), the numbers of holes of the shape, and what parts of it are connected to what. However, the most abstract notion of giving a set ##X## a topology, is that of determining what subsets of ##X## are open subsets. Can I get these two to correspond to each other? I.e. explain in what sense determining open subsets correspond to giving a set of points it's shape as described above? If not in the most general case, is this possible for topological manifold (that looks like R^n)?