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