Topology

Topology is a branch of advanced mathematics that focuses on sets and the manipulation and mapping of sets. General topology was traditionally subdivided into:

• continuous topology
• geometric topology

More recently the subject topology is divided into:

• algebraic topology
• point set topology

Topology also has a specific mathematical definition as a collection of open sets in a topological space.

In topology, a genus of a surface is the greatest number of distinct, continuous closed curves that may be drawn on it without separating the surface into distinct regions. The closed curves cannot be self-intersecting. The genus of the surface of a sphere is 0, while the genus of a doughnut shape is 1.

If you draw a circle on a sphere, then you can put a point A inside the circle and a point B outside the circle. The only way to connect A to B would have to cross the circle. So drawing the circle divides the surface of the sphere into two regions. But you can draw a circle around the edge (rim) of a doughnut without creating two regions. You can get from point A on one side of the line to point B on the other side, by going through the hole in the center.