Math 6510 • Differentiable Manifolds

Let \(X\) be a path connected, locally path connected, semi-locally simply
connected Hausdorff space. One of the main theorems of covering space theory
asserts that there is a simply connected space \(\widetilde{X}\) and a covering
map \(p:\widetilde{X}\to X\). Outline the construction of \(\widetilde{X}\) and \(p\)
as follows:
\begin{enumerate}
\item[(a)] Describe \(\widetilde{X}\) as a set.
\item[(b)] Describe the topology on \(\widetilde{X}\).
\item[(c)] Describe \(p\). You do not need to prove that \(p\) is a covering map.
\item[(d)] Outline the proof that \(\widetilde{X}\) is simply connected.
\end{enumerate}