a *rubber-sheet* topological space
[…] These manifolds have the metric property which allows different points to be related as locations, whether they be points on a number line, or infinite dimensional points in a Hilbert space, or points that support a boundary in a *rubber-sheet* topological space.
A manifold is defined by the types of relations between its points. But recently, category theory has expanded the geometric sub-structure of mathematical spaces from points to collections of more complex objects such as tangent lines (an example of a Sheaf) and even to collections of all possible algebraic relations supported by algebraic objects (an example of a topos or scheme). The general idea is to build structures not from sets but from mappings between undefined objects.