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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-strip.
This Video was produces for a topology seminar at the Leibniz Universitaet Hannover.
http://www-ifm.math.uni-hannover.de/~fugru/?topologie_teil1
Length: 83
Rating: 4.50 (192 ratings)
Tags: advent calendar geometry topology mathematics moebius strip klein bottle glue bothmer fugru cg
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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 make a torus, this shows that going first around through the hole and then along the outside is homeomorphic to going first along the outside and then through the hole.
Since these two path generate the fundamental group of the torus this proves that this group is abelan. q.e.d.
Remark: This is a very special property. Many topological spaces have nonabelian fundamental groups.
This video was produces for a topology seminar at the Leibniz Universitaet Hannover.
http://www-ifm.math.uni-hannover.de/~fugru/?topologie_teil1
Length: 93
Rating: 4.60 (34 ratings)
Tags: topology math fundamental group torus abelian path homotopy bothmer fugru cg
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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 mathematical tool to do this is called the Seifert-van Kampen Theorem.
This Video was produces for a topology seminar at the Leibniz Universitaet Hannover.
http://www-ifm.math.uni-hannover.de/~fugru/?topologie_teil1
Length: 54
Rating: 4.70 (15 ratings)
Tags: topology mathematics glueing glue torus seifert van kampen bothmer fugru cg
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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: The sphere is compact. Every infinite series on the sphere must therefore contain an accumulation point. Indeed this is the case for the preimage of points on a line.
Caution: This video does prove that the sphere is compact. For this one would have to consider ALL infinte sequences, not just one.
Remark2: The equally spaced points on the line are not shown.
(artistic freedom...)
This Video was produces for a topology seminar at the Leibniz Universitaet Hannover.
http://www-ifm.math.uni-hannover.de/~fugru/?topologie_teil1
Length: 117
Rating: 4.60 (19 ratings)
Tags: topology mathematics compact stereographic projection infinity bothmer fugru cg
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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 set and X itself and satisfies the two above properties (finite intersections und arbitrary unions stay in T) is called a topoloy on X. A space X with a topology T is called a topological space.
This Video was produces for a topology seminar at the Leibniz Universitaet Hannover.
http://www-ifm.math.uni-hannover.de/~fugru/?topologie_teil1
Length: 44
Rating: 4.50 (11 ratings)
Tags: topology mathematics open set union intersection bothmer fugru cg
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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.
http://www-ifm.math.uni-hannover.de/~fugru/?topologie_teil1
Length: 49
Rating: 4.00 (6 ratings)
Tags: topology mathematics homotopy path bothmer fugru cg
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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.
This Video was produces for a topology seminar at the Leibniz Universitaet Hannover.
http://www-ifm.math.uni-hannover.de/~fugru/?topologie_teil1
Length: 39
Rating: 4.30 (6 ratings)
Tags: topology mathematics homeomorphism fire bothmer fugru
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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 trivial fundamental group, i.e. it does not admit any non trivial closed paths it is called the universal covering. Here we see how a closed path on the circle is lifed to a non closed path on the spiral. Indeed the infinite spiral is the universal covering of the circle.
The name universal comes from an other property: The universal covering is a covering of every other covering. This is shown here with the 2:1 and the 3:1 covering of the circle. The universal covering covers both of them, but the 3:1 does not cover the 2:1.
From this universality property it follows also that every topological space has a unique universal covering. (not shown)
This Video was produces for a topology seminar at the Leibniz Universitaet Hannover.
http://www-ifm.math.uni-hannover.de/~fugru/?topologie_teil1
Length: 140
Rating: 4.80 (13 ratings)
Tags: topology mathematics universal covering circle spiral fundamental group bothmer fugru cg
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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 an endpoint can be joined to form a new path. This will become the addition law of the fundamental group.
The trivial path that goes from one point to the same point without ever moving can be added to a path without changing it. This will become the neutral element of the fundamental group.
If we have a path we can consider its direction (red circles moving). If we add the same path with opposite orientation to we obtain a path that is homotopic to the trivial path. This reverse path will become the inverse element in the fundamental group.
To obtain a true group we chose a fixed basepoint and consider only homotopy classes of paths that start and end there (not shown). This way we can add any two elements of the group.
This Video was produces for a topology seminar at the Leibniz Universitaet Hannover.
http://www-ifm.math.uni-hannover.de/~fugru/?topologie_teil1
Length: 113
Rating: 4.30 (14 ratings)
Tags: topology mathematics fundamental group path addition neutral element inverse bothmer fugru cg
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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 remaining pieces would then still be homeomorphic. On the other hand the first one has two components while the second one is still connected. Since connectedness is preserved by bi-continous maps we obtain a contradiction. Therefore a bi-continous map from the line to the circle can not exist. q.e.d.
This Video was produces for a topology seminar at the Leibniz Universitaet Hannover.
http://www-ifm.math.uni-hannover.de/~fugru/?topologie_teil1
Length: 24
Rating: 5.00 (2 ratings)
Tags: topology mathematics homeomorphy conectedness homeomorphism path bothmer fugru cg
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