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\vspace{-1in}
\begin{center}
 \large\bf \m Math 365,  Test \# 1
$\left.\hspace{2em}\oval(80, 40)\right/ 100$
 \end{center}
\begin{flushright}
October 2,  1996\\
Instructor: Bei Hu
\end{flushright}

\hspace{.5\textwidth} Name:\hrulefill

For a proof (Problems 3--7), please quote the theorem you use for each step.
\vspace*{5ex}

1. (32 points). True or False. (No proof or counter example needed).

(a). The countable union of countable sets is a countable set.
\vspace{.5in}

(b). The complex field is not an ordered field.
\vspace{.5in}

(c). If $\{G_n\}$ is open, then $\dis \bigcap_{n=1}^\infty G_n$ is open.
\vspace{.5in}

(d). If $E$ is a nonempty bounded set in $Q$ (the rationals), then
$\sup E$ exists and is also a rational.
\vspace{.5in}

(e). If $G$ is open, then $G$ is not closed.
\vspace{.5in}

(f). The product of two irrationals is always irrational.
\vspace{.5in}

(g). If $a$ is rational, $b$ is irrational, then $a+9b$ is always
irrational.
\vspace{.5in}

(h). Let $X$ be a metric space with a distance function $d(p,q)$,
then any neighborhood must contain infinitely many points.
\vspace{.5in}

\newpage
2. (20 points).  

(a). State PRECISELY the definition of a compact set.
\vspace{2in}

(b). State PRECISELY the definition of i) a metric, and ii) a metric space.
\vspace{2in}

(c). State PRECISELY the definition for a limit point for a set $E$.
\vspace{2in}

(d). State PRECISELY the definition for a perfect set.
\vspace{2in}
\newpage

3. (10 points). If $x$ and $y$ are complex numbers, prove that
$\Big||x|-|y|\Big| \le |x-y|$.
\vspace*{4in}


4. (10 points). 

(i). Do $E$ and $\bar{E}$ always have the same interiors?  Prove or give
a counter example.

(ii). Do $E$ and $E^\circ$ always have the same closure?  Prove or give
a counter example.


\newpage


5. (10 points). Suppose that $X$ is a metric space with $d$.
Suppose that $\{x_n\}$ is a sequence in $X$ such that 
$$
 \lim_{n\rightarrow \infty } x_n  = p.
$$
(We say $\displaystyle \lim_{n\rightarrow \infty } x_n = p$ if
for any neighborhood $N$ of $p$, there is a $N>0$ such that
$x_n\in N$ for all $n>N$.)

Let $$E = \{x_n\}_{n=1}^\infty \cup \{p\}. $$

(i) Is $E$ a closed set? Prove or give
a counter example.

(ii) Is $E$ a compact set? Prove or give
a counter example.
\newpage

6. (8 points).  Please explain:

(a) We know that the interval $(0,1)$ of the real line
is not a compact set. However, here is a ``proof''.

{\bf Proof:} We know that $[0,1]$ is a closed, compact set. 
$(0,1)$ is a subset of $[0,1]$. Therefore by Theorem, 
the set $(0,1)$, being a subset of a compact set,
is also a compact set.

What is wrong with this proof?

\vspace*{3.0in}

(b) We know that the set of real numbers in (0,1) is not
countable. However, here is a ``proof''.

{\bf Proof:} We write each number as $0.q_1q_2q_3\cdots$.
Each $q_j$ takes the value $0,1,2,3,4,5,6,7,8$ or $9$.
Thus $q_j$ has $10$ possible values. Now if we let
$q_j$ to be all these possible values, all real numbers
in (0,1) are taken care of. There are only countably number
of digits. Therefore, the set (0,1), as a countable union
of finite sets, is countable.

What is wrong with this proof?
\newpage


7. (10 points).  Let $J=\{1,2,3,\cdots\}$. Let ${\cal E}$ be the set
of all subsets of $J$. Prove that ${\cal E}$ is uncountable.

(The elements of ${\cal E}$ are subsets of $J$, such as:
$\{1,3,8,9, 99\}$, $\{1,3,5,7,9,\cdots\}$, $\cdots$) 
 
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