Mathematica command MoleculePlot3D["Caffeine"] 		This summer, we did not talk much about simply connected ness. A space is simply connected, if you can pull together any closed loop to a point within the space. A coffee cup is not simply connected if it has a handle because you can put a loop through the handle and you are unable to pull it together. There is a mathematical notion of "genus" which counts the number of holes. Mathematically, it is the first Betti number which an be computed using linear algebra. It does not count 3 dimensional holes like if you take a peach and remove the middle stone. While an annular region 1 ≤ x2+y2 &le 2 is not simply connected as the genus is 1, the three dimensional shell 1 ≤ x2+y2 &le 2 is simply connected. A nice joke in XKCD illustrates a bit also the culture differences in different fields. It is a very smart cartoon because it illustrates also the ambiguity in definitions. This field of topology was traditionally quite vague. The Caffeine (IN mathematica just type MoleculePlot3D["Caffeine"] has two holes. Its genus is 2. Since it is connected the number of connected components b0 = 1. The Euler characteristic is b0 - b1 = 1-2 =-1.