|
the klein bottle
The Klein Bottle is a surface on which you can move from outside to inside without crossing an edge. This shows that inside and outside are not universal concepts.
In this movie Klein's Bottle is constructed by gluing an rectangle along the edges. Then the bottle is cut up again to yield a Moebius
|
|
the fundamental group of the torus is abelian
This video illustrates the proof of the Theorem in the title. The proof goes like this:
Consider a rectangle. Then the path going up the left side of the rectangle and then along the top is homeomorphic to the path going first along the bottom and then up the right side.
Gluing the rectancle to
|
|
gluing a torus
Gluing is a good method to construct new topological spaces from known ones. Here a rectangles is glued along the edges to form a torus.
Often the fundamental group of the glued object can be calculated from the pieces (here a rectangles) and the glue (here two intersecting circles). The mathematic
|
|
compactness and stereographic projection
Here the stereographic projection of the sphere to the plane is illustrated. Also a proof that the plane is not compact is shown:
Proof: Consider equally spaced points along a line. This is an infinite sequence without an accumulation point. This can not happen on a compact set. q.e.d.
Remark 1: T
|
|
open sets
This video illustrates operations on open sets (symbolized by burning objects). The fire symbolizes that open sets have no border. The union of two open sets is again open. The intersection of to open sets is also open.
Let X be any set. Every collection T of subsets of X that contains the empty s
|
|
not a homeomorphism
Here an operation on a rectangle is shown that is not a homeomorphism. Punching a hole in a topological space is not bi - continuous.
After the hole is made the further change is a homeomorphism. The burning endges symbolize that the topological space depicted does not have an edge around the hole.
|
|
null-homotopic paths
A path is called null homotopic, if it can be contracted to a point.
This Video was produces for a topology seminar at the Leibniz Universitaet Hannover.
www - ifm.math.uni - hann over.de/ ~fugru/ ?topologie _ teil 1
|
|
coverings of the circle
A covering of a topological space X is a topological space Y together with a continuous surjective map from X to Y that is locally bi - continuos.
The infinite spiral is for example a covering of the circle. Notice how every path on the circle can be lifted to the spiral.
If a covering has a trivi
|
|
the fundamental group
This video illustrated the construction of the fundamental Group of a topological space.
Inside a topological space (symbolized by the brown box) we consider paths. Paths that are homotopic are considered equal (symbolized by the wiggeling of the paths).
Now two paths that share a beginning and a
|
|
non-homeomorphic topological spaces
This clip shows two non homeomorphic topological spaces (a line segment and a circle).
Proof: We have to show that there is no bi - continuos map from the line segment to the circle. If there was such a map we could remove a point of the line segment and the image of this point on the circle. The rem
|
|
< Prev 1 2Next > |
