# A covering space question for the “line with two origins”

I am learning about covering spaces for the first time and I came across the following problem connected to the subject:

Let $L$ be the line with two origins, which is the quotient space obtained by taking the disjoint union of two copies of $\mathbb{R}$ and identifying all equal pairs of nonzero points in the two copies. Construct a simply connected covering of $L$, and compute the fundamental group of $L$.

I am not yet comfortable with constructions in this area, so I was wondering if anyone visiting could walk me through such a construction. From what I understand, covering spaces can make the task of computing fundamental groups for several kinds of spaces quite manageable. I might be in error here, but I think simply connected spaces have trivial fundamental group. Will this factor into the computation of the fundamental group of $L$ (perhaps not to the point that the fundamental group of X is trivial, but still...)?

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This was asked and answered on MathOverflow: mathoverflow.net/questions/25472/… – NKS Jan 31 '12 at 5:09