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\begin{minipage}{4in}
\begin{center} 
{\large {\bf Homework 6 }} \\
Topology, Math 441, Spring 2020 \\
Topic: Subspace and Product Topology \\
{\bf Due:} Wednesday April 1st, 2020 \\
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\ \\

{\bf Reading:} Pages 46-56 from Chapter 4. \\

{\bf Problems:}

\begin{enumerate}
\item Let $Y\subset X$ have the subspace 
topology. 
\begin{enumerate}
\item Prove that if $A$ is open in $Y$ and $Y$ is open in $X$ then $A$
  is open in $X$.
\item Prove that if $A$ is closed in $Y$  and $Y$ is closed in $X$ then $A$
  is closed in $X$.
\end{enumerate}

\item Let $Y=(0,4] \cup \{5\} \subset \R$. Which of the following subsets of 
Y are open or closed in the subspace topology on $Y$ ? \\
\begin{inparaenum}
\item $(0,1)$
\item $(0,1]$
\item $\{1\}$
\item $(1,2)$
\item $(1,4]$
\item $[1,4]$
\item $\{5\}$
\item $\{4,5\}$

\end{inparaenum}



\item Let $X$ be a Hausdorff space and $Y \subset X$. Prove that 
the subspace topology on $Y$ is Hausdorff. 

\item Let $B\subset A \subset X$. Then $x \in A$ is a limit point of $B$ in $A$ 
if and only if $x$ is a limit point of $B$ in $X$. 

\item Prove that $X$ is Hausdorff if and only if the subset $\Delta(X)=\{(x,x)\ |\ x \in X\ \}$ is a closed subset of $X \times X$. 


\item Prove that a product of finitely many Hausdorff spaces is Hausdorff 
(Hint: Prove for two spaces and use induction.)


\item Let $\R_{{\rm fc}}$ denote $\R$ with the finite complement
  topology. Is the finite complement topology on $\R^2$ same as
  $\R_{{\rm fc}} \times \R_{{\rm fc}}$ ?

%\item Give an example to show that the quotient space of a Hausdorff
%  space need not be Hausdorff.

\end{enumerate}



{\bf Handin:} 1a, 2, 4, 5



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