Let $X$ be the topological space consisting of the standard 2-sphere together with a line segment from the north pole to the south pole. Compute $\pi_{1}(X)$ and construct the universal covering space of $X$.
By van Kampen theorem, this figure can be decomposed into a circle and a sphere. Then the fundamental group should be $\mathbb{Z}$. But I have no idea of universal covering space of this space.
You're right, according to van Kampen's theorem, this figure can be decomposed into a circle and a sphere. The fundamental group should therefore be $\mathbb{Z}$. To construct the universal cover space of $X$, you can imagine the original space as an ordinary sphere $S$ sitting in a 3-dimensional Euclidean space, and imagine the line $L$ connecting the two poles as being drawn on the outside of the sphere (where it's easier for me to see). The universal cover space of this is an infinite number of spheres (in both directions), so that the North Pole of the ith sphere is stuck to the South Pole of the (i + 1)th sphere. The coverage map simply maps each sphere downwards. Inserting a line segment between each sphere gives the correct answer.
