\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 \# 3}

\msk

\ctln{{\bf Due: }Sept. 25}

\bsk


\item{11.} Let $(X,{\Cal T})$ be a topological space, $B\subseteq X$ a subset, and 
${\Cal T}_B$ the subspace topology on $B$. If $A\subseteq B$, show that the subspace
topology that it inherits from $B$ is the \u{same} as the subspace topology that it
inherits from $X$.

\bsk

\item{12.} Show that if $A\subseteq X$ and $(X,{\Cal T})$ is Hausdorff, then the
subspace topology on $A$ is Hausdorff.

\bsk

\item{13.} Show that if $(X,d)$ and $(Y,d^\prime)$ are metric spaces, then the product
topology on $X\times Y$ is metrizable. [There are lots of (correct) choices of metric on 
$X\times Y$; you can take your cue from ${\Bbb R}^2$ .]

\bsk

\item{14.} Show that if $(X,{\Cal T}), (Y,{\Cal T}^\prime)$ are topological spaces and $x_0\in X$, then
the function 

\ssk

\ctln{${\iota}_{x_0} : Y\rightarrow X\times Y$ , ${\iota}_{x_0}(y) = (x_0,y)$}

\ssk

\item{} is continuous.

\bsk

\item{15.} Show that if $(X,d)$ is a metric space, then the metric $d : X\times X\rightarrow 
{\Bbb R}$ is continuous (where $X\times X$ has the product topology). Show, further, that
the metric topology ${\Cal T}$ is the \u{coarsest} topology on $X$ for which $d$ is continuous.

\ssk

\item{} (Hint: show that if ${\Cal T}^\prime\subsetneq {\Cal T}$, then $N_d(x_0,\epsilon)\notin{\Cal T}^\prime$
for some $x_0$ and $\epsilon$ ; now look at problem \# 14.)

\bsk

\item{16.} Show that, \u{if} $X$ is an infinite set, then the finite complement topology ${\Cal T}_f$ on 
$X\times X$ is \u{not} a product topology, i.e., there do not exist topologies 
${\Cal T},{\Cal T}^\prime$ on $X$ whose product topology is ${\Cal T}_f$. On the other hand, 
if $X$ is finite, show that ${\Cal T}_f$ on $X\times X$ \u{is} a product topology.

\ssk

\item{} (Hint: the \u{basis} for the product topology would have to be $\subseteq {\Cal T}_f$ ...)



\vfill
\end