
\documentstyle[12pt]{article}
\setlength{\topmargin}{-0.75in}
\setlength{\oddsidemargin}{-.250in}
\setlength{\textwidth}{7.0in}
\setlength{\textheight}{9.5in}
\parindent 0pt

\def\R{{\bf R}}
\def\Q{{\bf Q}}


%
\begin{document}

\def\u{\vskip 10pt}
\def\h{\hskip 10pt}
\def\w{\vskip 28pt}
\def\v{\vskip 38pt}

22M:132: Topology Exam 1

Sept 30, 2008

\u

[10]~~ 1.)  Definition:  The point $x \in X$ is a limit point of $A$ if 

\vfill

[12]~~ 2.)  Suppose $\R$ = the set of real numbers has the standard topology.  Let $\Q$ = the set of rational numbers.  
Calculate the following in $\R$:

\u\u

${\Q}^{\hskip 2pt o} = \underline{\hskip 1in}$
\hfill
$\overline{\Q} = \underline{\hskip 1in}$
\hfill
${\Q}{\hskip 2pt '} = \underline{\hskip 1in}$

\u\u


3.)  The following two statements are false.  Show that the statements are false by providing counter-examples.  You do not 
need to explain your counter-examples.
\u

[9]~ 3a.)  If $x_n \in A$, then there exists a unique point $x \in \overline{A}$ such that $x_n 
\rightarrow x$.

\vfill



[9]~ 3b.)  If $f:  X \rightarrow Y$ is continuous, then $\overline{f(A)} \subset f(\overline{A})$.

\vfill




\eject



[60]~~ Prove 2 of the following 3.  Clearly indicate your choices.

\begin{enumerate}



 \vspace{.15in}

\item  The product of two Hausdorff spaces is Hausdorff.
 
 \vspace{.15in}

\item Let $Y$ be a subspace of $X$.  If $A \subset Y$, then $Cl_Y(A) = \overline{A} \cap Y$.

 \vspace{.15in}

  \item  Show that $D( (x_1, y_1), (x_2, y_2) ) = min\{|x_1 - x_2|, \frac{|y_1 - y_2|}{2}  \}$ is a metric 
on $\R^2$ where 
$(x_i, y_i) \in \R^2$.  Show that this metric generates the standard topology on $\R^2$.


\end{enumerate}

\end{document}
