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\lhead{Mathematics 468}
\rhead{J.~Palmieri}

\begin{center}
{\large \textbf{Exam 1}} \\
\textbf{March 4, 1998}
\end{center}

General instructions: When I say ``$X$ is a topological space,'' I
mean that $X$ is a na\"{\i}ve topological space: $X$ is a subset of
$\mathbf{R}^{n}$ for some $n$.  

\begin{questions}

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\uplevel{\textbf{Definitions.}}

\question[5]
Let $X$ be a topological space and let $A$ be a subset of $X$.  Define
what it means for a point $y$ of $X$ to be a \emph{limit point} of
$A$.

\question[5]
Define the term \emph{connected}, as applied to topological spaces.

\question[5]
Let $X$ and $Y$ be topological spaces, and 
let $f : X \xrightarrow{} Y$ be a function.  What does it mean for $f$
to be \emph{continuous}?

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\uplevel{\textbf{True or false:} For the next two problems, tell me
whether each statement is true or false.  If it's true, give a
\emph{brief} reason why.  If it's false, give a brief reason or a
counterexample.}

\question[10]
The circle $S^{1}$, defined by
\[
S^{1} = \{ (x, y) \in \mathbf{R}^{2} \ : \  x^{2} + y^{2} = 1 \},
\]
is homeomorphic to the open interval $(0,3)$.

\question[10]
Let $f: X \xrightarrow{} Y$ be a function between
topological spaces $X$ and $Y$.  Then $f^{-1}(Y) = X$.

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\uplevel{\textbf{Types of sets:} For the next question, you may
give reasons if you want, but they are optional.  (If you get the
answers right without any justification, you \emph{will} get full
credit.)}

\question
Let $X$ be the topological space 
\[
X = [0,1] \cup (2,3] \cup \{4\} \cup 
\left( \rule{0in}{0.2in} (6,7) \cap \mathbf{Q} \right),
\]
viewed as a subset of $\mathbf{R}$.  Let $A = (2,3]$.  Let $B =
[0,\sqrt{40}) \cap X$.  Let $C=(6,7) \cap \mathbf{Q}$.

\begin{parts}
\part[5] Is $A$ open as a subset of $X$?

\part[5] Is $A$ compact?

\part[5] Is $A$ connected?

\part[5] Is $B$ open as a subset of $X$?

\part[5] Is $B$ closed as a subset of $X$?

\part[5] Is $B$ connected?

\part[5] Is $C$ open as a subset of $X$?

\part[5] Is $C$ open as a subset of $\mathbf{R}$?

\part[5] Is $4$ a limit point of $X$?

\end{parts}


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\uplevel{\textbf{Theory:} Do \emph{two} of the following four
problems.}

\question[10]
Let $X$ be a topological space and let $A$ be a subset of $X$.  Show
that if $A$ contains all of its limit points, then $A$ is closed.

\question[10]
Let $X$ be a topological space and let $A$ be a subset of $X$.  Show
that if $A$ is closed, then $A$ contains all of its limit points.

\question[10]
Prove that the interval $[0,1]$ is connected.

\question[10]
If $A$ and $B$ are countable sets, show that $A \cup B$ is countable,
also.  Then use this to explain why the set of irrational numbers is
uncountable.

\end{questions}

\end{document}
