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

Category:

Howto & Style

Tags:

• topology
• mathematics
• universal
• covering
• circle
• spiral
• fundamental
• group
• bothmer
• fugru
• cg

[4jeremy9]

makes me want to wrap myself in cellophane to see my shape! :D....... no but seriously these animations are great models. Can u use any other shape to cover a sphere without having to use "infinity" AND avoid closing a path? or would the closest to this be a fractal?

[bothmer]

@4jeremy9: This is a very good question. The generalization of "no closed path" to two dimension is "every path can be shrunk to a point on the surface" (as in our video "Null-homotopic Paths"). With this generalisation the sphere is its own universal covering space! For other surfaces more complicated things can occur.

[exiledninja0]

this is better than eating shrooms

[starkey7uk]

That's very good! Nicely illustrated.

[cantormath]

Nice work...

[exodd22]

odd...

[mychemroxmisox42]

thats interesting

[trouvel]

scary very scary

[kelemengabi]

very good .

[leporidus]

I bekomme richtig Bock, mal wieder mit Povray 'rumzuspielen. :D