\input amstex

\magnification=1200

\def\ctln{\centerline}
\def\msk{\medskip}
\def\bsk{\bigskip}
\def\ssk{\smallskip}
\def\ra{\rightarrow}

\UseAMSsymbols

\ctln{\bf Math 871 Problem Set 1}

\msk

Starred (*) problems are due Thursday, Sept. 11.

\bsk

\item{6.} For a set $X$, and (fixed) $a\in X$, the \underbar{excluded} \underbar{point} \underbar{topology}
on $X$ is the collection of subsets ${\Cal T}_a = \{{\Cal U}\subseteq X\ :\ a\not\in{\Cal U}\}\cup\{X\}$.
Show that ${\Cal T}_a$ is a topology on $X$.

\msk

\item{(*)} 7. Show that the set 
${\Cal T} = \{(a,\infty)\ :\ a\in{\Bbb R}\}\ \cup\ \{[a,\infty)\ :\ a\in{\Bbb R}\}\ \cup\ \{\emptyset,{\Bbb R}\}$
of subsets of $\Bbb R$ is a topology on $\Bbb R$.

\ssk

\item{} [Note that ${\Cal T}$ can be thought of as the sets ${\Cal U}\subseteq{\Bbb R}$ so that whenever
$a\in{\Cal U}$ and $b\geq a$, then $b\in{\Cal U}$.]

\msk

\item{8.} (a) Show that if $f:{\Bbb R}\ra{\Bbb R}$ is an increasing function [$a\geq b\ \Rightarrow f(a)\geq f(b)$],
then, as a function from $({\Bbb R},{\Cal T})$ to $({\Bbb R},{\Cal T})$, $f$ is continuous.

\msk

\item{} (b) Show, conversely, that if  $f:({\Bbb R},{\Cal T})\ra({\Bbb R},{\Cal T})$ is continuous, then $f$ is increasing!

\item{} [Consequently, this topology captures increasing functions precisely as its continuous functions...]

\item{} [Hint: suppose $f$ \underbar{isn't} increasing: show it is \underbar{not} continuous.]

\msk

\item{(*)} 9. [Munkres, p.83, \#13.3] (a) Show that for any set $X$, the sets

\hfill $\{{\Cal U}\subseteq X\ :\ X\setminus{\Cal U}\ \text{is}\ \text{countable}\}\cup\{\emptyset\}$ form a topology
on $X$.

\ssk

\item{} (b) Do the sets $\{{\Cal U}\subseteq X\ :\ X\setminus{\Cal U}\ \text{is}\ \text{infinite}\}\cup\{\emptyset\}$
always form a topology on $X$ ? Explain why or why not.

\msk

\item{10.} [Munkres, p.83, \#13.4(a)] If ${\Cal T}_\alpha$ are all topologies on the same set $X$, 
show that $\bigcap_\alpha{\Cal T}_\alpha$ (the intersection of all of the topologies) is 
also a topology on $X$. Is $\bigcup_\alpha{\Cal T}_\alpha$ (their union) a topology on $X$?

\msk

\item{(*)} 11. With the topology ${\Cal T}$ on $\Bbb R$ from problem \#6, and ${\Cal T}^\prime$ the ``usual''
topology on ${\Bbb R}$ (open sets are unions of neighborhoods), 
show that every continuous function 

\item{} $f:({\Bbb R},{\Cal T})\ra ({\Bbb R},{\Cal T}^\prime)$ must be \underbar{constant}.

\ssk

\item{} [Hint: Suppose not! Show that $f$ \underbar{can't} be continuous.]

\msk

\item{12.} With the topologies from problem \#11, show that there do exist
continuous functions $f:({\Bbb R},{\Cal T}^\prime)\ra ({\Bbb R},{\Cal T})$ which are \underbar{not} constant.

\ssk

\item{} [How could $f^{-1}([a,\infty))$ and $f^{-1}((a,\infty))$ both be open?]

\vfill
\end
