Difference between revisions of "Topology"
(Explaining "genus number" with examples) |
m (links) |
||
| Line 9: | Line 9: | ||
*point set topology | *point set topology | ||
| − | Topology also has a specific mathematical definition as a collection of open | + | Topology also has a specific mathematical definition as a collection of [[open set]]s in a [[topological space]]. |
| − | In topology, a ''genus'' of a surface is the greatest number of distinct, continuous closed | + | In topology, a ''genus'' of a surface is the greatest number of distinct, [[continuous]] [[closed_set|closed]] [[curve]]s 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. | 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. | ||
[[category:mathematics]] | [[category:mathematics]] | ||
Revision as of 18:33, May 24, 2008
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.