In this sense, topology is a vital arbiter in the “discrete vs continuous” wars. The very existence of topology as a discipline shows that “discrete properties” always exist even if you only work with continuous objects. On the other hand, topology always assumes that these features are “derived” – they’re some of the properties of objects and these objects are deeper and that may have many other, continuous properties, too. The topological, discrete properties of these objects are just projections or caricatures of the “whole truth”.

— Euler characteristic

— Lubos Motl

2013.06.02 Sunday ACHK