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  \phantom{}\\[-2\baselineskip]
  We now check that $p$ is continuous. It is
  enough to check that $p^{-1}(U)$ is open for $U
  \in \mathcal{U}$. If $[\alpha] \in p^{-1}(U)$,
  then $[\alpha] \in ([\alpha], U) \subset
  p^{-1}(U)$, so $p^{-1}(U)$ is open.

  To see that $p$ is a \gls{covmap}, we first show
  that each
  \[ p|_{([\alpha], U)} : ([\alpha], U) \to U \]
  is a homeomorphism. As $U$ is \gls{pathconn},
  for any $y \in U$ there is a path $\gamma$ in
  $U$ from $\alpha(1)$ to $y$, so $p([\alpha
  \concat \gamma]) = y$, so the map is surjective.
  If $[\beta], [\beta'] \in ([\alpha], U)$ maps to
  the same thing under $p|_{([\alpha], U)}$, then
  $\beta$ and $\beta'$ end at the same point. So
  there exist \glspl{path} $\gamma$, $\gamma'$ in
  $U$ such that $[\beta] = [\alpha \concat
  \gamma]$, $[\beta'] = [\alpha \concat \gamma']$.
  \begin{center}
    \includegraphics[width=0.6\linewidth]{images/4581ba67e7c84e3c.png}
  \end{center}
  So
  \[ [\beta']
  = [\alpha \concat \gamma \concat \invpath\gamma
  \concat \gamma'] = [\alpha \concat \gamma] =
  [\beta] \]
  where the first equality comes from the fact
  that $\invpath\gamma \concat \gamma'$ is a loop
  in $U$, so \gls{homotopic} to a constant loop in
  $X$. So $p|_{([\alpha], U)}$ is injective. So
  $p|_{([\alpha], U)}$ is a bijection, and
  continuous. It is also open as
  \[ p(([\beta], V)) = V ,\]
  so $p|_{([\alpha], U)}$ is a homeomorphism. We
  now claim that $p^{-1}(U)$ is partitioned into
  $([\alpha], U)$s. When checking that $p$ is
  continuous, we saw that they cover, so need to
  show that if they intersect then they are equal.
  So let $[\gamma] \in ([\alpha], U) \cap
  ([\beta], U)$, i.e. there are \glspl{path}
  $\alpha'$, $\beta'$ in $U$ such that $[\gamma] =
  [\alpha \concat \alpha'] = [\beta \concat
  \beta']$. Let $[\delta] \in ([\alpha], U)$ So
  \[ [\delta] = [\alpha \concat \alpha''] =
  [\alpha \concat \alpha' \concat
  \invpath{(\alpha')} \concat \alpha''] = [\beta
  \concat \ub{\beta' \concat \invpath{(\alpha')}
  \concat \alpha''}_{\subset U}] \]
  so $[\delta] \in ([\beta], U)$
  \begin{center}
    \includegraphics[width=0.6\linewidth]{images/a1ff9e405d9b4003.png}
  \end{center}
  i.e. $([\alpha], U) \subseteq ([\beta], U)$,
  similarly for the opposite containment.

  Finally, we need to show that $\tilde{X}$ is
  \gls{simpconn}. Note: if $\gamma : I \to X$ is a
  \gls{path}, then its \gls{lift} $\tilde{\gamma}
  : I \to \tilde{X}$ startng at
  $[\constpath{x_0}]$ ends at $[\gamma]$, because
  \[ s \mapsto [t \mapsto \gamma(st)] : I \to
  \tilde{X} \]
  is the \gls{lift}. So if a loop $\gamma$ in $X$
  \glspl{lift} to a loop in $\tilde{X}$ based at
  $[\constpath{x_0}]$, so
  \[ [\gamma] = [\constpath{x_0}] \implies \ps p
  \pio\bs(\tilde{X}, [\constpath{x_0}]) = \{e\}
  \subseteq \pio\bs(X, x_0) .\]
  But $\ps p$ is injective, so
  \[ \pio\bs(\tilde{X}, [\constpath{x_0}]) = \{e\} \qedhere \]
\end{proof}

\subsection{The Galois Correspondence}

If $p : \tilde{X} \to X$ is a \gls{covmap},
$\tilde{X}$ \gls{pathconn}, $x_0 \in X$ and
$\tilde{x}_0 \in p^{-1}(x_0)$, then
\[ \ps p : \pio\bs(\tilde{X}, \tilde{x}_0) \to
\pio\bs(X, x_0) \]
is injective, giving a subgroup of $\pio\bs(X,
x_0)$.

If $\tilde{x}_0' \in p^{-1}(x_0)$ is another
basepoint, let $\gamma$ be a \gls{path} in
$\tilde{X}$ from $\tilde{x}_0$ to $\tilde{x}_0'$.
Then $p \circ \gamma$ is a loop based at $x_0$,
and we have
\[ [p \circ \gamma]^{-1} \concat \ps p
\pio\bs(\tilde{X}, \tilde{x}_0) \concat [p \circ
\gamma] = \ps p \pio\bs(\tilde{X}, \tilde{x}_0')
\le \pio\bs(X, x_0) \]
So fixing a \gls{pathconn} based space $\bs(X,
x_0)$ we get
\begin{align*}
  \left\{\substack{
    \text{based \gls{pathconn} \glspl{covmap}} \\
    \text{$p : \bs(\tilde{X}, \tilde{x}_0) \to
    \bs(X, x_0)$}
  }\right\}
  &\stackrel{p \mapsto \Im(\ps
  p)}{\xrightarrow{\hspace*{2cm}}}
  \left\{\substack{
    \text{subgroups of $\pio\bs(X, x_0)$}
  }\right\} \\
  \left\{\substack{
    \text{\gls{pathconn} \gls{covmap}} \\
    \text{$p : \tilde{X} \to X$}
  }\right\}
  &\stackrel{p \mapsto \Im(\ps
  p)}{\xrightarrow{\hspace*{2cm}}}
  \left\{\substack{
    \text{conjugacy classes of} \\
    \text{subgroups of $\pio\bs(X, x_0)$}
  }\right\}
\end{align*}

\begin{flashcard}[galois-surj-prop]
\begin{proposition*}[Surjectivity of Galois
  correspondence]
  \cloze{Let $X$ be \gls{pathconn}, \gls{locpathconn},
  and \gls{slsc}. Then for any $H \le \pio\bs(X,
  x_0)$ there is a $p : \bs(\tilde{X},
  \tilde{x}_0) \to \bs(X, x_0)$ with $\ps p
  \pio\bs(\tilde{X}, \tilde{x}_0) = H$.}
\end{proposition*}

\begin{proof}
  Let $\ol{X} \stackrel{q}{\to} X$ be the
  \gls{unicov} we have constructed in
  \nameref{unicov_existence}. Define $\sim_H$ on
  $\ol{X}$ by $[\gamma] \sim_H [\gamma'] \iff
  \gamma(1) = \gamma'(1)$ and $[\gamma \concat
  \invpath{(\gamma')}] \in H \le \pio\bs(X, x_0)$.
  So:
  \begin{enumerate}[(i)]
    \item $[\gamma] \sim_H [\gamma]$.
    \item If $[\gamma] \sim_H [\gamma']$ then
      $[\gamma \concat \invpath{(\gamma')}] \in
      H$, so $[\gamma' \concat
      \invpath{(\gamma')}] \in H$. So $[\gamma']
      \sim_H [\gamma]$.
    \item If $[\gamma] \sim_H [\gamma']$,
      $[\gamma'] \sim_H [\gamma'']$, then
      $\gamma(1) = \gamma'(1) = \gamma''(1)$, and
      \[ [\gamma \concat \invpath{(\gamma'')}] =
      [\gamma \concat \invpath{(\gamma')} \concat
      \gamma' \concat \invpath{(\gamma'')}]
      = \ub{[\gamma \concat
      \invpath{(\gamma')}]}_{\in H}
      \ub{[\gamma' \concat
      \invpath{(\gamma'')}]}_{\in H}
      \in H .\]
      So $[\gamma] \sim_H [\gamma'']$.
  \end{enumerate}
  So $\sim_H$ is an equivalence relation. Define
  $\ol{X}_H = \ol{X} / \sim_H$, the quotient
  space, and $p_H : \ol{X}_H \to X$ to be the
  induced map. If $[\gamma] \in ([\alpha], U)$,
  $[\gamma'] \in ([\beta], U)$
  satisfy $[\gamma] \sim_H [\gamma']$ then
  $([\alpha], U)$ and $[\beta], U)$ are identified
  by $\sim_H$, as $[\gamma \concat \eta] \sim_H
  [\gamma' \concat \eta]$ for any \gls{path}
  $\eta$ in $U$. So $p_H$ is a \gls{covmap}.

  It remains to show that $\ps{(p_H)}
  \pio\bs(\ol{X}_H, [[\constpath{x_0}]]) = H \le
  \pio\bs(X, x_0)$. If $[\gamma] \in H$ then the
  \gls{lift} of $\gamma$ to $\ol{X}$ starting at
  $[\constpath{x_0}]$ ends at $[\gamma]$, so the
  \gls{lift} to $\ol{X}_H$ ends at $[[\gamma]] =
  [[\constpath{x_0}]]$, so is a loop, i.e.
  \[ H \subseteq \ps{(p_H)} \pio\bs(\ol{X}_H,
  [[\constpath{x_0}]) .\]
  If $[\gamma] \in \ps{(p_H)} \pio\bs(\ol{X}_H,
  [[\constpath{x_0}]])$ then the \gls{lift}
  $\ol{\gamma}$ of $\gamma$ to $\ol{X}$ starting
  at $[\constpath{x_0}]$ ends at $[\gamma]$, so
  $[\gamma] \sim_H [\constpath{x_0}]$, as it
  becomes a loop in $\ol{X}_H$ by assumption. So
  $[\gamma] \in H$.
\end{proof}
\end{flashcard}
