\input amstex

\magnification=1200

\def\ctln{\centerline}
\def\msk{\medskip}
\def\bsk{\bigskip}
\def\ssk{\smallskip}
\def\u{\underbar}

\UseAMSsymbols

\ctln{\bf Math 970 Homework \# 4}

\msk

\ctln{{\bf Due: }Oct. 2}

\bsk

\item{17.} For $A,B\subseteq X$ with $(X,{\Cal T})$ a topological space, 
if $A$ is open in $X$ and $B$ is closed in $X$, 
then $A\setminus B$ is open and $B\setminus A$ is closed.

\bsk

\item{18.} Show that if $A,B\subseteq X$, then

\msk

(a) $\overline{A\cup B} = \overline{A}\cup\overline{B}$

\ssk

(b) $\overline{A\cap B} \subseteq \overline{A}\cap\overline{B}$, but that equality does not hold in general,

\ssk

(c) $\overline{A\setminus B} \supseteq \overline{A}\setminus\overline{B}$, but that equality does not hold in general.

\bsk

\item{19.} Show that if $A\subseteq X$ and $X$ has two topologies ${\Cal T}\subseteq {\Cal T}^\prime$, then 
if $x\in X$ is a limit point of $A$ w.r.t. ${\Cal T}^\prime$, then it is a limit point of $A$ w.r.t. ${\Cal T}$.

\bsk

\item{20.}Show that if $A_i\subseteq X_i$ for all $i\in I$, then

\msk

\ctln{$\displaystyle \overline{\prod_i A_i}$= $\displaystyle \prod_i \overline{A_i}$ $\subseteq$ $\displaystyle \prod_i X_i$}

for \underbar{both} the product and box topologies.



\bsk

\item{21.}Find the closure of the set $(0,1)\subseteq{\Bbb R}$, when ${\Bbb R}$
has the 

\msk

(a) finite complement topology

\ssk

(b) infinite (open) ray to the right topology

\ssk

(c) discrete topology

\ssk

(d) {\it lower limit topology}, generated by the basis ${\Cal B} = \{[a,b) : a,b\in {\Bbb R}\}$


\vfill\end

