Let .

Since X is semi-locally simply connected, there exists aGalois correspondencebetween subgroups of the fundamental group and path-connected covering spaces of X.

The fundamental group of X is .

We shall find all the subgroups of and connect each subgroup to its corresponding path-connected covering space of X.

("A subgroup of " "A corresponding path-connected covering space of X")

1. {e} . This is a universal cover of X.

2. .

3. .

4. .

5. , where n and m are positive integers greater than 1.

In this case, consider a covering map given by , where is a polar coordinate in the plane . The map , where n is a positive integer, wraps the circle around itself n times. Thus is a covering space of , corresponding a fundamental group .

6. , where n is a positive integer greater than 1. It is similar to (5).

7. itself.

This problem becomes much more difficult if . It is because free product groups are involved in the fundamental group of X, and it is not easy to find all subgroups of a free product group and draw its Cayley graph.