The figure 8 is two circles tangent at the base point. It's fundamental group is the free group on 2 generators. Click here for more details. The first generator moves around the top loop counterclockwise, while its inverse moves around the top loop clockwise. The second generator accesses the bottom loop.
The descending earring is a generalization of the figure 8. Infinitely many circles are tangent at a point, and the fundamental group includes the free group on infinitely many generators. The nth generator in the group winds around the nth circle counterclockwise, and its inverse implies a clockwise rotation.
However, there are more homotopy classes, beyond those described by the aforementioned free group. Suppose the loops have circumferences of 1/2, 1/4, 1/8, 1/16, 1/32, and so on. I think you can see that a path of length 1, running at constant speed, can wind around all the circles and come to rest at the origin. This path is continuous everywhere, even at its end. This loop is not homotopic to any of the paths described by our free group.
A complete description of the fundamental group of the descending earring is interesting, but beyond the scope of this page. For now, let's return to the world of covering spaces.
Every open set about the origin contains some circles. Travel around one of these circles and find a loop that is not homotopic to a point. Therefore S is not semilocally simply connected. In the next section we'll see that a locally simply connected space always has a universal cover. Since the descending earring is not locally simply connected, it might not have a universal cover. In fact it doesn't, and we're going to prove that now.
In the previous section we built a universal cover for n circles tangent at a point. Can we extend this construction to infinite dimensions? That would be a universal cover for S. Indeed, we can build the fractal, with its continuous projection onto S, but the origin is not evenly covered. Every open set containing the origin includes some circles, yet the preimage has no loops. The two open sets cannot be homeomorphic, and our constructed fractal is not a covering space.
Let's build a few covering spaces for S, even though they might not be universal.
Look again at the covering space we constructed for the figure 8. One of the two circles was broken open, and covered by the real line. The other circle was replicated, and hung off of each integer. We're going to do the same thing for the descending earring.
Let C1 be the real line, with a descending earring hanging off of each integer. Every time a path moves around the outer circle of S, its lift advances one unit along the real line. when the path moves around any of the inner circles, its lift moves around the corresponding circles in the descending earring that is associated with the current integer. One circle is broken open and the rest of S is copied again and again.
To build C2, start with the fractal in 2 dimensions that acts as a universal cover for two tangent circles, and hang a descending earring off of each intersection. When a path moves around the first or second circle its lift travels along line segments in the fractal. When the path moves along any of the inner circles, its lift moves around the corresponding circles in the descending earring that is associated with the current intersection.
Build C3 by hanging earrings from the intersections in the 3 dimensional fractal, and so on. Verify that each Cn is a covering space for S.
Now suppose there is a universal cover U for S. Remember that U contains a small open neighborhood that is homeomorphic to an open neighborhood of S. Let this neighborhood contain all the circles of radius 1/n and beyond.
Let Cn cover S, as described above. Look at the preimage of the nth circle, in U, and in Cn. These preimages are a loop and an interval respectively. The morphism from U to Cn, if it exists, is continuous. Restrict this function to the loop in U, and its image must be an interval in T. Enclose either of its endpoints in a small open set in T, and the preimage is not open in U. The function is not continuous, and we have a contradiction. Hence there is no universal cover for the descending earring.