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\ctln{\bf Problems from Munkres assigned Fall 2004, of ``qual'' caliber:}

\medskip



p.118 \#\ 10(a): Let $A$ be a set, $\{X_\alpha\}_{\alpha \in J}$
a family of topological spaces, and $\{f_\alpha\}_{\alpha \in J}$
a family of functions $f_\alpha:A \rightarrow X_\alpha$.  Show that
there is a unique coarsest topology on $A$ relative to which
 each of the $f_\alpha$ is continuous.



\msk



p.111 \#\  8: Let $f,g:X \rightarrow {\Bbb R}$ be continuous.
Let $h:X \rightarrow {\Bbb R}$ be defined by $h(x)=\min\{f(x),g(x)\}$
 for all $x \in X$.  Show that $h$ is continuous.



\msk


p.118 \#\ 3:  Show that the product of Hausdorff spaces is Hausdorff.



\msk



p.101 \#\  8: Let $A_\alpha$ be subsets of a topological
space $X$. 
Determine whether $\overline{\cap A_\alpha}=
\cap \overline{A_\alpha}$ and 
if not, determine whether
 either containment $\subset$ or $\supset$ holds.



\msk



p.152 \#\  11: Let $p:X \rightarrow Y$ be a quotient map
 such that $p^{-1}(\{y\})$ is connected for each $y \in Y$. 
Show that if $Y$ is connected, then so is $X$.



\msk



p.158 \#\  8(b): If $A \subset X$ and $A$ is path connected,
is $\overline{A}$ necessarily path connected?



\msk



p.171 \#\  5: Show that if $A,B$ are disjoint compact subsets of the Hausdorff space $X$, then there
are disjoint open sets ${\Cal U},{\Cal V}\subseteq X$ with $A\subseteq {\Cal U}$ , $B\subseteq{\Cal V}$ .



\msk




p.199 \#\  5: Let $Y$ be a Hausdorff space and let
 $f,g:X \rightarrow Y$ be continuous.  
Show that $\{x \mid f(x)=g(x)\}$ is closed in $X$.



\msk



p.218 \#\  3: Let $X$ be a compact Hausdorff space.  Show that
 $X$ is metrizable iff $X$ has a countable basis.



\msk



p.330 \#\  3(b): Show that a contractible space is path connected.



\msk



p.335 \#\  4: Let $a_0 \in A \subset X$ and let
$r:X \rightarrow A$ be a retracion.  
Show that 
$r_*:\pi_1(X,a_0) \rightarrow \pi_1(A,a_0)$ is surjective.




\msk



p.341 \#\  6(a): Let $p:E \rightarrow B$ be a covering map.
 Show that if $B$ is Hausdorff, then so is $E$.



\msk



p.353 \#\  1: Show that if $A$ is a retract of $B^2$, then
 every continuous map $f:A \rightarrow A$ has a fixed point.



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%p. \#\  : 








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\ctln{\bf Problems from Hatcher so far assigned, of ``qual'' caliber:}

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p.39 \#\ 13 : If $x_0\subseteq A\subseteq X$ with $A$ path-connected, then the inclusion-induced map 
$\pi_1(A,x_0)\rightarrow \pi_1(X,x_0)$ is surjective 
$\Leftrightarrow$ 
every path in $X$ with endpoints in $A$  is homotopic, rel endpoints, to a path in $A$.



\msk



p.39, \#\ 14 : Show that $\pi_1(X\times Y)\cong \pi_1(X)\times \pi_1(Y)$



\msk



p.40, \#\ 20 : If $f_t: X\rightarrow X$ is a homotopy from the identity map to 
itself, then 
$t\mapsto f_t(x_0)$ is an element in the center of $\pi_1(X,x_0)$ .



\msk



p.53, \#\ 6 : Attaching an $n$-cell, $n\geq 3$ to a space does not change its 
$\pi_1$ .



\msk



p.53, \#\ 7: Compute the fundamental group of the space obtained from $S^2$ by identifying 
the north and south poles.



\msk



p.53, \#\ 8: Compute the fundmental group of the space obtained by gluing the curves 
$S^1\times\{x_0\}$ in two copies of a 2-torus together.



\msk



p.54, \#\ 14 : Compute the fundamental group of the space obtained from the cube by identifying 
opposite faces, each with a 90-degree right-hand twist.



\msk



p.79, \#\ 1 : If $p:\widetilde{X}\rightarrow X$ is a covering map, and $A\subseteq X$,
then $p: p^{-1}(A)\rightarrow A$ is also a covering map.



\msk



p.79, \#\ 3: If $p:\widetilde{X}\rightarrow X$ is a finite-sheeted covering, then $\widetilde{X}$
is compact and Hausdorff $\Leftrightarrow$ $X$ is.



\msk



p.79, \#\ 9: If $X$ is path-connected and locally path-connected, with 
$\pi_1(X)$ finite, then every map $f: X\rightarrow S^1$ is null-homotopic.



\msk



p.80, \#\ 15: If $p:\widetilde{X}\rightarrow X$ is a universal covering, $A\subseteq X$ 
path-connected
and locally path-connected, and $\widetilde{A}$ a path-component 
of $p^{-1}(A)$, then $p:\widetilde{A}\rightarrow A$ is the covering space corresponding 
to the kernel
 of the map $i_*:\pi_1(A)\rightarrow \pi_1(X)$ .



\msk



p.86, \#\ 2: If $A\subseteq X$ are connected graphs, then there is a retraction $r:X\rightarrow A$ . 


\noindent [Maybe it's enough to do this for finite graphs?]


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