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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 (mathematics)|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 [[torus]] (doughnut shape) is 1.  

In topology, a ''[[genus (mathematics)|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 [[torus]] (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 torus 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 torus 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.  

A way of thinking about topology (albeit a non-rigorous one) is that two shapes are topologically the same, if they can be transformed into one another if one can be turned into the other by stretching it (or parts of it), but there must be no cutting or gluing of the shape. If you imagine the objects are made of clay then you can stretch or compress the clay or pull it around, but you cannot tear it at any point.

A way of thinking about topology (albeit a non-rigorous one) is that two shapes are topologically the same, if they can be transformed into one another if one can be turned into the other by stretching it (or parts of it), but there must be no cutting or gluing of the shape. If you imagine the objects are made of clay then you can stretch or compress the clay or pull it around, but you cannot tear it at any point.

[[category:mathematics]][[Category:Topology]]

[[Category:Mathematics]][[Category:Topology]]