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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:

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]]