How do I conclude that $|\pi_1(X,x_0):p_*(\pi_1(C,c_0))|$ is the number of sheets of $p$?

Question

Here is a theorem in Hatcher's algebraic topology.

(Hatcher-Algebraic Topology p.61)

Let $(X,x_0),(C,c_0)$ be topological spaces and $p:(C,c_0)\rightarrow (X,x_0)$ be a covering map.

If $C$ and $X$ are path-conneted, then $|\pi_1(X,x_0):p_*(\pi_1(C,c_0))|$ is the number of sheets of $p$.

So far, I have proven below statement

(Munkres-Topology p.346)

If $C$ is path-connected then $|\pi_1(X,x_0):p_*(\pi_1(C,c_0))|=|p^{-1}(x_0)|$.

I haven't used path-connectivity of $X$ yet. How do I prove Hatcher's theorem via the statement I have proved?

Answer 1

Prove that for any $x_0 \in X$, the set $\{ x \in X : |p^{-1}(x)| = |p^{-1}(x_0)| \}$ is clopen in $X$. It follows that if $X$ is connected, this is the whole set.

Now it's a problem of definitions: the number of sheets of $p$ is defined to be the cardinality $\kappa$ such that $|p^{-1}(x)| = \kappa$ for every $x$. It doesn't always exist: sometimes the cardinalities are not the same everywhere.

By the way I think you missed this part of p.61 of Hatcher's book:

If $p : \tilde X \to X$ is a covering space, then the cardinality of the set $p^{-1}(x)$ is locally constant over $X$. Hence if $X$ is connected, this cardinality is constant as $x$ ranges over all of $X$. It is called the number of sheets of the covering.