Difference between revisions of "Topology Glossary"
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Revision as of 18:18, 6 July 2011
This page provides an introduction to many topological terms and concepts.
Closed sets contain all of their boundaries. If a set contains no boundaries, then this is satisfied by default.
A surface is a 2 dimensional manifold.
Orientability is a property of manifolds that describes whether directions, or orientations, can be consistently defined in the surface. Directions can be consistently defined on an orientable manifold, but not on a non-orientable manifold.
In the case of surfaces, a 2 mandifold is orientable if every closed path preserves the orientation of an entity that travels along it. A 2 mandifold is non-orientable if there exists at least one closed path that reverses the orientation of something traveling along it.
For instance, say that rotation in the counter-clockwise direction is considered to be positive, and clockwise, negative. If an object spinning counter-clockwise moves along a closed path in an orientable surface, it will return to its starting point spinning counter-clockwise. Yet, if the surface is non-orientable, then it will twist in such a way that, on certain paths, the object will return spinning clockwise, though the motion of its rotation never changed. Equivalently, in non-orientable surfaces, an object within the surface can travel along a path that will lead it back to its start point, but with its right and left sides flipped.
A good demonstration of non-orientability on the Mobius Strip is visible on this page from Plus Magazine. When viewing this, remember that the face is within the surface, not merely pasted on one of the sides.
Orientability and non-orientability are most often discussed in relation to surfaces. Nevertheless, the properties are descriptive of higher dimensional manifolds as well. One can consider a 3 dimensional space where certain paths would lead back to their start points, but flip the right and left sides of things that traveled along them.
Here is a more technical explanation: In a non-orientable manifold, there exists at least one path around the manifold such that, if we take a set of basis vectors for the manifold and move them along this path, they will arrive at their starting point with the following result. The determinant of the matrix composed of these basis vectors will have the opposite sign (positive/ negative) when it arrives back at the starting point as it did when it left.
Not all image pages will discuss in depth how non-orientability arises for a particular surface; the phenomenon is discussed, however, for the Mobius Strip and Real Projective Plane, with images to explain the phenominon.
- Massey, William. (1991). A Basic Course in Algebraic Topology (Graduate Texts in Mathematics). New York: Springer-Verlag.