\magnification1200
\magnification1200
\def\pf{\medskip\noindent{\sl Proof:}\ }
%%%%%%%%
\noindent{\bf Prob. 1 \& 3, p. 221.}
\item{1.} Let $X$ be a CW complex and $A\subset X$
a compact subset.
\itemitem{(a)}
Show that $A$ intersects only finitely many open cells of $X$.
Where do you use ``closure{--}finiteness''?
\pf
To help with this problem as well as part (b) to come, put a
partial ordering on the open cells of $X$ as follows:
$e_\alpha\prec e_\beta$ provided
$\bar e_\beta\cap e_\alpha\neq\emptyset$ and
$\dim e_\alpha < \dim e_\beta$.
Write $e_\alpha< e_\beta$ provided there exits a sequence
of cells $e_{\alpha_0}=e_\alpha$, $e_{\alpha_1}$, \dots
$e_{\alpha_r}=e_\beta$ such that $e_{\alpha_i}\prec e_{\alpha_{i+1}}$
for $0\leq i\dim e_\beta$, ${\cal I}(e_\beta,r)=\emptyset$.
The set of open cells $e_\alpha$ with $e_\alpha