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\begin{document}

\title[Homework \#1]%
{{\large \quad MTH3400 --- Geometry 1 --- Spring 1997}\\[0.2in]
{\bfseries Homework \#1}}


%\author{Alexander I.~Suciu}

\maketitle

\begin{enumerate}

\item Let $X$ be a locally Euclidean space.
  \begin{enumerate}
    \item Show that each component of $X$ is an open subset of $X$. 
    \item Show that, if $X$ is connected, then $X$ is path-connected.
  \end{enumerate}

\vskip 0.15truein

\item Let $X$ and $Y$ be connected, locally Euclidean spaces
of the same dimension.  If $f:X\rightarrow Y$ 
is bijective and continuous, show that $f$ is a homeomorphism. 

\vskip 0.15truein

\item  Let $\pi_{\pm}: S^n\setminus {p_{\pm}}\to  \R^n$ 
be the stereographic projections.  Write down explicit 
formulas for these maps, and prove that they are homeomorphisms.  


\vskip 0.15truein

\item Let $D^n=\{ v\in \R^n \mid ||v||\le 1\}$ be the unit $n$-disk, 
with boundary $\partial D^n = S^{n-1}$.  Prove that $D^n/S^{n-1}$ 
is homeomorphic to $S^n$.  

\vskip 0.15truein

\item  Define an equivalence relation $\sim$  on 
$S^n =\{ v\in \R^{n+1} \mid ||v||= 1\}$ by writing 
$v \sim w$ if and only if $v = \pm w$.  The quotient space 
$\RP^n = S^n/\sim$ is called the (real) projective $n$-space. 
Let $\pi:S^n\to \RP^n$, $\pi(v)=\{\pm v\}$ be the canonical 
projection. For each $i$, $1\le i \le n+1$, define a 
subset $U_i$ of $\RP^n$ by
\[
U_i=\{\pi(x^1,\dots, x^{n+1}) \mid x^i \ne 0\}.
\]
Prove the following facts, which together show that 
$\RP^n$ is an $n$-manifold.
  \begin{enumerate}
    \item $U_i$ is open in $\RP^n$.
    \item $\{ U_1,\dots,U_{n+1}\}$ covers $\RP^n$. 
    \item There is a homeomorphism $\phi_i: U_i \to \R^n$.  
    \item $\RP^n$ is compact, connected, and Hausdorff.  
  \end{enumerate}


\vskip 0.15truein

\item  Let $X$ be an $n$-dimensional manifold with boundary.  Prove: 
  \begin{enumerate}
    \item $\partial X \cap \Int(X) = \emptyset$. 
    \item $\partial X $ is an $(n-1)$-manifold. 
    \item $\Int X $ is an $n$-manifold. 
  \end{enumerate}


\vskip 0.15truein

\item  Let $M$, $N$ be manifolds with boundary.  
Prove that $\partial (M\times N) = \partial M \times N \cup 
M \times \partial N$. 

\vskip 0.15truein

\item  Let $V$ and $W$ be real vector spaces.  Show how a choice of bases
$e_1,\ldots,e_m$ for $V$ and $f_1,\ldots,f_n$ for $W$ defines an isomorphism 
of $\Hom{}_{\R}(V,W)$ with the real vector space of $n\times  m$ 
matrices with real coefficients,  $\Mat{}_{n\times m}({\R})$. 
Shows that this isomorphism transforms composition
%
\begin{align*}
\Hom{}_{\R}(V,W)\times \Hom{}_{\R}(T,V) &\to 
\Hom{}_{\R}(T,V) \\
										(f,g) &\mapsto f\circ g
\end{align*}
%
\noindent into matrix multiplication
%
\begin{align*}
\Mat{}_{n\times m}({\R})\times \Mat{}_{m\times k}({\R}) 
		&\to \Mat{}_{n\times k}({\R}) \\
										(A,B) &\mapsto A\cdot B
\end{align*}


\end{enumerate}

\end{document}