Exercise 60.3

Question

Let $p\colon E 	o X$ be the map constructed in the proof of Lemma $60.5$. Let $E'$ be the subspace of $E$ that is the union of the $x$-axis and the $y$-axis. Show that $p |_{E'}$ is not a covering map.

Amit Awasthi · Accepted Answer

\begin{proof}
  Consider the base point $x_0$, the center of the figure-eight $X$. A
  neighborhood $U 
i x_0$ contains the union $V$ of open intervals in $A$ and
  $B$ which intersect in exactly $\{x_0\}$. $(p |_{E'})^{-1}(V)$ is then equal
  to the union of open intervals around integers on the $x$-axis and open
  intervals around integers on the $y$-axis. But none of these intervals are
  homeomorphic to $V$, since removing an integer in an interval gives two
  connected components, while removing its image $x_0$ in $V$ gives four connected components.
\end{proof}

Exercise 60.3

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Exercise 60.3

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