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\begin{center}\begin{large}\textbf{Math 101 - Problem Set 10\\
Due Thursday, November 30}\end{large}\end{center}


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\begin{enumerate} 

\item Consider the following topologies on $\R$:

\begin{itemize}
\item $\T_1=$ the standard topology,
\item $\T_2 = $ the cofinite topology,
\item $\T_3 =$ the topology with basis $\{ (-\infty, a ) \mid a \in \R \}$,
\item $\T_4 = $ the topology with basis $\{ (a, b) \mid a < b \textrm{ and } a, b \in \Z \},$
\item $\T_5 = $ the lower limit topology (which has basis $\{[a,b) \mid a < b\}$).
\end{itemize}

We already showed in class that $\T_5$ is finer than $\T_1$, i.e. $\T_1 \subseteq \T_5$. For the remaining pairs of topologies, either show that one is finer than the other or that they are not comparable. Be sure to justify your answer.

\item Let $X$ be a set and $\B$ a basis. Let $\T$ be the topology generated by $\B$. Show that $\T$ is the smallest (i.e. coarsest) topology containing $\B$. That is, if $\T'$ is another topology on $X$ and $\B \subseteq \T'$, then $\T \subseteq \T'$.

\item Consider the topology $\T_K$ on $\R$, defined in class, that is generated by the following basis:
$$\B = \{ (a,b) \mid a <b\} \cup \{ (a,b) - K \mid a <b\},$$ where $K = \{\frac{1}{n} \mid n \in \Zp\}$.

\begin{enumerate}
\item Show that $\T_K$ and the lower limit topology are not comparable.
\item Are $\T_K$ and the upper limit topology comparable? If so, which is finer? (The upper limit topology has basis $\{(a,b] \mid a < b\}$.)
\end{enumerate}

\item \textit{I know we've talked about this before a few times, but there is still some confusion and I just want to make sure it's clear:}
\vspace{.2cm}

Let $X$ be a set and $\T$ a topology on $X$. Suppose that if $x\in X$ then $\{x\} \in \T$. Show that $\T$ is the discrete topology.

\vspace{.5cm}
\noindent \textbf{{The next two problems depend on material that we will cover on Tuesday, November 28.}}

\item Show that the order topology on $\Z$ is the discrete topology.

\item Consider $\R_{\geq 0} \times \R_{\geq 0}$ given the order topology (with the dictionary order). In each case, show whether or not the given subset is open.

\begin{enumerate}

\item $U = [1,2] \times (1,2)$
\item $U = (1,2) \times (1,2)$
\item $U = (1,2) \times [1,2]$
\item $U = [0,1) \times [0,\infty)$


\end{enumerate}

\end{enumerate}
\end{document}