Exercises:

1. Explain why the map $\phi: \mathbb{C} - \{0, 1/2\}\longrightarrow \mathbb{C} - \{-1/4\}$ given by $\phi(z) = z(z-1)$ is not a covering projection?
2. Show that the map $f : S^1 \longrightarrow S^1$ given by $f(z) = z^k$ is a covering projection for every $k \in \mathbb{N}$.
3. Suppose $p: {\tilde X}\longrightarrow X$ is a covering projection and $E$ is a closed subset of $X$. Is the map
   $$p: {\tilde X} - p^{-1}(E)\longrightarrow X - E$$
   a covering projection?
4. Find a discrete subset $E$ of $\mathbb{C}$ such that $\sin: \mathbb{C} - E \longrightarrow \mathbb{C} - \{-1, 1\}$ is a covering projection.
5. Suppose that $p: {\tilde X}\longrightarrow X$ and $q:{\tilde Y}\longrightarrow Y$ are covering projections then the product map $(p, q): {\tilde X}\times{\tilde Y} \longrightarrow X\times Y$ given by
   $$(p, q)(z, w) = (p(z), q(w)),\quad z \in \tilde X, w \in \tilde Y,$$
   is a covering projection. In particular the plane $\mathbb{R}^2$ is a covering space of the torus $S^1\times S^1$.
6. Let $Y$ be the infinite grid
   $$Y = \{(x, y) \in \mathbb{R}^2/ x \in \mathbb{Z}$$    or $$y \in \mathbb{Z}\}$$
   is a covering projection of the figure eight loop. Draw the figure eight loop on the torus.
7. Show that the set $G$ in theorem (15.2) is closed without using the Hausdorff assumption on $T$.

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Lecture XVI - Lifting of paths and homotopies

In the last lecture we discussed the lifting problem and proved that the lift if it exists is uniquely determined by its value at one point. In this lecture we shall prove the important result that covering projections enjoy the path lifting and covering homotopy properties. This theorem is fundamental in the the theory of covering projections and will be used in the next lecture to define an action of the fundamental group on the fibers.