\input amstex

\magnification=1200

\def\u{\underbar}
\def\ctln{\centerline}
\def\ni{\noindent}
\def\ssk{\smallskip}
\def\msk{\medskip}
\def\bsk{\bigskip}
\def\ra{\rightarrow}
\def\itt{\item}
\def\pr{\hskip-10pt}
\def\hsk{\hskip10pt}
\def\hhsk{\hskip20pt}
\def\lra{$\Leftrightarrow\ $ }
\def\exs{$\exists\ $}
\def\foa{$\forall\ $}
\def\cau{{\Cal U} }
\def\cav{{\Cal V}}
\def\sset{\subseteq}
\def\finv{f^{-1}}
\def\catm{${\Cal T}\ $}
\def\cat{{\Cal T}}
\def\catpm{$\cat^\prime\ $}
\def\catp{\cat^\prime}
\def\top{topology}
\def\tops{topologies}
\def\cts{continuous}
\def\smin{\setminus}
\def\cab{{\Cal B}}
\def\cabm{${\Cal B}\ $}
\def\cas{{\Cal S}}
\def\casm{${\Cal S}\ $}
\def\bbbr{{\Bbb R}}
\def\bbbn{{\Bbb N}}
\def\bbbq{{\Bbb Q}}

\UseAMSsymbols

\ctln{\bf Math 970 Homework \# 8}

\msk

\ctln{{\bf Due: November 11} }

\bsk


\item{33.} Show that if $(X,{\Cal T})$ is compact, and ${\Cal T}^\prime\sset{\Cal T}$, then
$(X,{\Cal T}^\prime)$ is compact.

\bsk

\item{34.} Show that if $(X,{\Cal T})$ is a topological space and $A,B$ are compact subsets
of $X$, then $A\cup B$ is compact.

\bsk

\item{35.} Give an example of a space $(X,{\Cal T})$ and subsets $A,B\sset X$ so that
$A$ and $B$ are compact but $A\cap B$ is not.

\ssk

\item{} (Note: your space $X$ cannot be Hausdorff....)

\bsk

\item{36.} Let $X$ = $\Bbb R$ with the infinite ray topology 

\ssk

\ctln{$\Cal T$ = $\{(a,\infty) : a\in {\Bbb R}\}\cup \{\emptyset,{\Bbb R}\}$}

\ssk

Show that $A$ = $\{0\}$ is a compact subset of $X$, but its closure $\bar{A}$ \u{isn't}.


\bsk

\item{37.} Show that if $(X,\cat)$ is a Hausdorff space and $A,B\sset X$ are disjoint compact
subsets of $X$, then there are subsets $\cau,\cav\in \cat$ so that $A\sset \cau$, $B\sset \cav$,
and $\cau\cap\cav =\emptyset$ .



\vfill
\end

