Topology (from the Greek τόπος, “place”, and λόγος, “study”), as a branch of mathematics, can be formally defined as "the study of qualitative properties of certain objects (called topological spaces) that are invariant under a certain kind of transformation (called a continuous map), especially those properties that are invariant under a certain kind of equivalence (called homeomorphism)." To put it more simply, topology is the study of continuity and connectivity.

All models below are made with TopMod. The parametric ones, are further processed either using ParaCloud GEM or Smart Duplicate from Ticket01. I used Maya for the renderings.
Some of them are available on my Shapeways shop.