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\ctln{\bf Math 970 Homework 9}

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\ctln{Due Thursday, Nov. 20}

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\item{38.} Show that if $X$ is limit point compact, and $A$ is a closed subset of $X$, then
$A$ is limit point compact.

\bsk

\item{39.} Give an example of a limit point compact space $X$ and a continuous function
$f: X\rightarrow Y$ for which $f(X)\subseteq Y$ is not limit point compact.

\bsk

\ni Note: for the purposes of the following problems, `` regular'' means 
points and (disjoint) closed sets can be separated with open sets, ``normal'' means 
disjoint closed sets can be separated, $T_3$ means $T_1$ and regular, and
$T_4$ means $T_1$ and normal.

\bsk

\item{40.} Show by examples that the continuous image of $T_3$ need not
be $T_3$, and that the continuous image of a non-$T_3$ space can be $T_3$!

\bsk

\item{41.} Show that every closed subset of a normal space is normal, and that every 
closed subset of a $T_4$ space is $T_4$.

\bsk

\item{42.} Show that for any collection $X_\alpha\neq \emptyset$ of topological spaces, if 
$\displaystyle\prod_\alpha X_\alpha$ is $T_4$ in the product topology, then $X_\alpha$ is 
$T_4$ for all $\alpha$.

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\item{} (Hint: embed $X_\alpha$ in $\displaystyle\prod_\alpha X_\alpha$ as a closed subset!)

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