Difference between revisions of "Topology"
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'''Topology''' is a branch of advanced [[mathematics]] that focuses on [[set]]s and the manipulation and [[mapping]] of sets. General topology was traditionally subdivided into: | '''Topology''' is a branch of advanced [[mathematics]] that focuses on [[set]]s and the manipulation and [[mapping]] of sets. General topology was traditionally subdivided into: | ||
| − | *continuous topology | + | *[[continuous topology]] |
| − | *geometric topology | + | *[[geometric topology]] |
More recently the subject topology is divided into: | More recently the subject topology is divided into: | ||
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Topology also has a specific mathematical definition as a collection of [[open set]]s in a [[topological space]]. The open sets can be manipulated, forming the [[basis]] of the topology. The opposite of open sets are [[closed set]]s. Topology may also be defined in terms of closed sets, but this yields an [[equivalence relation|equivalent]] definition. | Topology also has a specific mathematical definition as a collection of [[open set]]s in a [[topological space]]. The open sets can be manipulated, forming the [[basis]] of the topology. The opposite of open sets are [[closed set]]s. Topology may also be defined in terms of closed sets, but this yields an [[equivalence relation|equivalent]] definition. | ||
| − | 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. | + | In topology, a ''[[genus (mathematics)]]'' 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:Topology]] | [[category:mathematics]][[Category:Topology]] | ||
Revision as of 01:41, August 31, 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:
More recently the subject topology is divided into:
Topology also has a specific mathematical definition as a collection of open sets in a topological space. The open sets can be manipulated, forming the basis of the topology. The opposite of open sets are closed sets. Topology may also be defined in terms of closed sets, but this yields an equivalent definition.
In topology, a genus (mathematics) 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.