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\newcommand{\myTitle}{HW 1, due Friday, February 8}
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\begin{enumerate}
  \item Prove the first clause of Prop.\ 1.7 in the topology handout.

    {\sc solution:}  [My first solution starts here.]

  \item Prove Prop.\ 1.9 in the topology handout.

    {\sc solution:}

  \item Give the simplest example of a topological space that is not Hausdorff.

    {\sc solution:}

  \item Let $X$ and $Y$ be disjoint copies of $\R$ with the usual topology.
    For $x\in X$ and $y\in Y$, say $x\sim y$ if $x=y\neq0$ as real numbers.
    Define $L = (X\cup Y)/\sim$ as a topological space with the quotient
    topology.  In other words, if
    \begin{align*}
      \pi\colon X\cup Y&\to L\\
      a&\mapsto[a],
    \end{align*}
    where $[a]=\{b\in X\cup Y: a\sim b\}$, then a subset $U\subseteq L$ is open
    if and only if $\pi^{-1}(U)$ is open in $X\cup Y$ (which means
    $\pi^{-1}(U)\cap X$ and $\pi^{-1}(U)\cap Y$ are open a subsets of the real
    numbers).  Thus, $L$ is essentially the real number line with two origins,
    $0_X$ from~$X$ and $0_Y$ from $Y$.
    Show that~$L$ is locally Euclidean but not Hausdorff. 

    {\sc solution:}

  \item Recall that $\proj^2$ is the set of one-dimensional vector subspaces of
    $\R^3$ and a {\em line} in $\proj^2$ is a two-dimensional subspace.  Prove
    that two points in $\proj^2$ determine a unique line and conversely, two
    lines determine a unique point.

    {\sc solution:}

  \item Consider $\proj^n$ with its standard atlas.  Calculate the transition
    function from $U_0$ to $U_1$, i.e., $\phi_1\circ\phi_0^{-1}$.

    {\sc solution:}

  \item Suppose $M$ is a manifold, let $f\colon M\to\R$, and let $(U,h)$ and
    $(V,k)$ be two charts about $p\in M$.  Show that
    if $f$ is differentiable at $p$ relative to a chart $(U,h)$, then  $f$ is
    differentiable at $p$ relative to $(V,k)$.

    {\sc solution:}

\item Let $V$ be an $n$-dimensional vector space over $\R$.  A choice
  of ordered basis $\mathbb{B}$, for $V$ gives a linear isomorphism
\[
h_{\mathbb{B}}\colon V\to\R^n.
\]
Give $V$ a topology by declaring a set open in $V$ iff its image under
$h_{\mathbb{B}}$ is open in $\R^n$.  Then $(V,h_{\mathbb{B}})$ is a
chart at each point in $V$, hence,
$\mathfrak{A}:=\{(V,h_{\mathbb{B}})\}$ is an atlas.  The maximal atlas
containing $\mathfrak{A}$ determines a differentiable structure on $V$
and makes $V$ a manifold.  Show that the choice of a different ordered
basis determines the same differentiable structure.  In this sense,
$V$ has a canonical manifold structure.

    {\sc solution:}

  \item Is $(-1,1)\subset\R$ diffeomorphic to $\R$?  Is the open disc of
    radius $1$ in $\R^2$ diffeomorphic to $\R^2$?  You may use the internet to
    help answer this question if you cannot figure it out on your own.

    {\sc solution:}

\end{enumerate}
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