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\begin{center}\begin{large}\textbf{Math 101 - Problem Set 9\\
Due Tuesday, November 21}\end{large}\end{center}


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\begin{enumerate} 

\item Determine whether the following subsets of $\R$ (given the standard topology) are open or not. Justify your answer.

\begin{enumerate}
\item $\R - \Z$.
\item $\Z$
\item $\displaystyle\bigcup_{0<a<b<1} [a,b]$
\item $\left\{\frac{1}{a} \mid  a\in \Z_+\right\}$

\end{enumerate}

\item Let $\mathcal{T}_1$ be the standard topology on $\R$ and $\mathcal{T}_2$ the cofinite topology on $\R$.
\begin{enumerate}
\item Show that if $U \in \mathcal{T}_2$, then $U \in \mathcal{T}_1$. i.e. $\mathcal{T}_2 \subseteq \mathcal{T}_1$. \\
(We say in this situation that $\mathcal{T}_1$ is \textbf{\textit{finer}} than $\mathcal{T}_2$ and that $\mathcal{T}_2$ is \textbf{\textit{coarser}} than $\mathcal{T}_1$.)
\item Give an example to show that the reverse inclusion doesn't hold.
\end{enumerate}

\item Determine whether each of the following collections of subsets is a basis for a topology on $\R$. Justify your answer. If it is a basis, describe the topology it generates.

\begin{enumerate}
\item $\B = \{ B \subset \R \mid B \textrm{ is finite} \}$
\item $\B = \{ B \subset \R \mid B \textrm{ is infinite} \}$
\item $\B = \{ (a, b) \mid a < b \textrm{ and } a, b \in \Z \}$
\item $\B = \{ \R - \{x\}  \mid x\in \R \}$
\item $\B = \{ (a-1, a+1) \mid a\in \R \}$
\end{enumerate}

\item Let $X$ be a set. Let $\mathcal{T}$ be the collection of all subsets $U$ of $X$ such that $X-U$ is either countable or is all of $X$. Show that $\mathcal{T}$ is a topology on $X$. 

\item Let $X$ be a topological space. Let $A \subseteq X$. Suppose that for each $x \in A$, there is an open set $U$ containing $x$ such that $U \subseteq A$. Show that $A$ is open in $X$.

\item Consider the set $X = \{a, b, c, d\}$. Come up with 8 topologies on $X$ that are different from each other even after permuting the elements. You can just circle the open sets like we did in class. You don't need to justify your answer. \textit{(Note that there are actually more possibilities than 8--you don't have to list the rest of them!)}



\end{enumerate}
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