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\section{Surfaces (7-8 lectures)}

\begin{flashcard}[topological-surface]
\begin{definition*}
    A \emph{topological surface} is a topological
    space $\Sigma$ such that
    \cloze{
    \begin{enumerate}[(a)]
        \item $\forall \rho \in \Sigma$, there is
            an open neighbourhood $p \in U \subset
            \Sigma$ such that $U$ is homeomorphic
            to $\RR^2$, or a disc $D^2 \subset
            \RR^2$, with its usual Euclidean
            topology.
        \item $\Sigma$ is Hausdorff and second
            countable (has a countable base)
    \end{enumerate}
    }
\end{definition*}
\end{flashcard}

\subsubsection*{Remarks}

\begin{enumerate}[(1)]
    \item $\RR^2 \cong D(0, 1) = \{x \in \RR^2
        \colon \|x\| < 1\}$
    \item A space $X$ is \emph{Hausdorff} if for
        $p \neq q$ in $X$, there exists
        \emph{disjoint} open sets with $p \in U$
        and $q \in V$ in $X$. A space is
        \emph{second countable} if it has a
        countable base, i.e. $\exists \{U_i\}_{i
        \in \NN}$ open sets such that every open
        set is a union of some of the $U_i$. Part
        (a) is the main point of the definition,
        and part (b) is technical honesty and
        convenience.
    \item If $X$ is Hausdorff / second countable
        so are subspaces of $X$. Euclidean space
        has these properties. (For second
        countable, consider open balls $B(c, r)$
        with $c \in \QQ^n \subset \RR^n$ and $r
        \in \QQ_+ \subset \RR_+$).
\end{enumerate}

\subsubsection*{Examples of Topological Surfaces}

\begin{enumerate}[(i)]
    \item $\RR^2$ the plane
    \item Any open set in $\RR^2$, i.e. $\RR^2
        \setminus Z$ where $Z$ is closed, for
        example $Z = \{0\}$, $\RR^2 \setminus
        \{0\}$ is a surface,
        \[ Z = \{(0, 0) \cup \{(0, Y_n) \colon n =
        1, 2, \dots\}\} \]
    \item Graphs: $f \colon \RR^2 \to \RR$
        continuous graph:
        \[ \Gamma_f = \{(x, y, f(x, y)) \colon (x,
        y) \in \RR^2\} \subset \RR^3 \]
        subspace topology. Then this is a surface.

        \myskip
Recall: if $X$, $Y$ topological spaces, then the
product topology on $X \times Y$ has basic open
sets $U \times V$ where $U \subset X$, $V \subset
Y$ open.

\myskip
Feature: $g \colon Z \to X \times Y$ is continuous
if and only if
\[ \pi_X \circ g \colon Z \to X \]
\[ \pi_Y \circ g \colon Z \to Y \]
are both continuous (where $\pi_X \colon X \times
Y \to X$ and $\pi_Y \colon X \times Y \to Y$ are
the canonical projections).

\myskip
Application:
$f \colon X \to Y$ continuous, $\Gamma_f \subset
X \times Y$ is homeomorphic to $X$, $S(x) = (x,
f(x))$ is continuous and then $\pi_X \mid_f$ and
$S$ are inverse homeomorphisms. So $\Gamma_f \cong
\RR^2$ for any $f \colon \RR^2 \to \RR$
continuous, so $\Gamma_f$ is a topological
surface.
    \item The sphere:
        \[ S^2 = \{(x, y, ) \in \RR^3 \colon x^2 +
        y^2 + z^2 = 1\} \]
        (subspace topology).
        \begin{center}
            \includegraphics[width=0.6\linewidth]
            {images/2b7f2a5e98b811ed.png}
        \end{center}
        \[ \pi_+ \colon S^2 \setminus \{0, 0, 1\}
        \to \RR^2 \quad ((z = 0) \subset \RR^3) \]
        \[ (x, y, z) \mapsto \left( \frac{x}{1 -
        z}, \frac{y}{1 - z} \right) \]
        $\pi_+$ is continuous and has an
        inverse
        \[ (u, v) \mapsto \left( \frac{2u}{u^2 +
        v^2 + 1}, \frac{2v}{u^2 + v^2 + 1},
        \frac{u^2 + v^2 - 1}{u^2 + v^2 + 1} \right) \]
        so $\pi_+$ is a continuous bijection
        with continuous inverse and hence a
        homeomorphism. Similarly:
        \begin{center}
            \includegraphics[width=0.6\linewidth]
            {images/97dee23498b811ed.png}
        \end{center}
        $\pi_i \colon S^2 \setminus \{(0, 0, -1)\}
        \to \RR^2\}$
        \[ (x, y, z) \mapsto \left( \frac{x}{1 +
        z}, \frac{y}{1 + z} \right) \]
        Stereographic projection from south pole,
        also a homeomorphism, so $S^2$ is a
        topological surface: the open sets $S^2
        \setminus \{(0, 0, 1)\}$ and $S^2
        \setminus \{0, 0, -1\}$ cover $S^2$ is
        Haudorff and second countable (inherited
        from $\RR^3$). Note: $S^2$ is compact.
    \item The \emph{real projective plane}. The
        group $\ZZ_2$ acts on $S^2$ by
        homeomorphisms via the antipodal map: $a
        \colon S^2 \to S^2$, $a(x, y, z) = (-x,
        -y, -z)$. $\ZZ_2 \injto
        \operatorname{Homeo}(S^2)$, non-trivial
        element $\to a$.

        \begin{definition*}[Real projective plane]
            The real projective plane is the
            quotient of $S^2$ by identifying every
            point with its antipodal image:
            \[ \mathbb{RP}^2 = S^2 / \ZZ_2 = S^2 /
            \sim \]
            where $x \sim a(x)$.
        \end{definition*}
\end{enumerate}