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\begin{center}\begin{large}\textbf{Math 131 - Problem Set 1\\
Due Tuesday, Sept 11}\end{large}\end{center}



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\begin{enumerate} 

\item Show that an infinite sequence of points $a_1, a_2, a_3, \ldots$ in a metric space has at most one limit point. 

\item
\begin{enumerate}
\item Show that any sequence in a metric space with a limit is a Cauchy sequence.
\item Show (with an example) that the converse of (a) is not true.
\end{enumerate}

\item Recall that a subset $U$ of a metric space $X$ is open iff for all $x \in U$, there is some $\epsilon >0$ such that $x \in B_\epsilon(x) \subset U$. Show that any open ball is in fact an open set.

\item Let $X$ be a metric space. Show that the collection of open sets on $X$ (given by the definition in the previous exercise) is a topology.

\item Consider the following distance functions on $\mathbb{R}^n$:
$$e((x_1,x_2, \ldots, x_n),(y_1, y_2, \ldots, y_n)) = \text{max}\{|y_i - x_i|\}$$ and $$f((x_1,x_2, \ldots, x_n),(y_1, y_2, \ldots, y_n)) = \sum |y_i - x_i|.$$
\begin{enumerate}
\item Show that $(\mathbb{R}^n, e)$ and $(\mathbb{R}^n, f)$ are metric spaces.
\item Show that $e$ and $f$ induce the same topology on $\mathbb{R}^n$ (i.e. a set is open in one iff it's open in the other).
\end{enumerate}

\item Let $X$ be a metric space. Show that a sequence $p_1, p_2, p_3, \ldots$ of points in $X$ has limit $p$ if and only if every open set containing $p$ contains all but finitely many of the $p_i$. (This gives a characterization of limits using open sets without explicitly using the metric.)

\item Let $X$ be a topological space, with basis $\mathcal{B}$ generating the topology $\mathcal{T}$. Show that $\mathcal{T}$ is the smallest topology on $X$ that contains $\mathcal{B}$. That is, every other topology containing $\mathcal{B}$ is finer than $\mathcal{T}$. (\textit{We'll define bases in class on Thursday, Sept 6}.)

\item Give an example of a topological space $X$ and an infinite sequence of bases $$\mathcal{B}_1 \supsetneq \mathcal{B}_2  \supsetneq  \mathcal{B}_3  \supsetneq \cdots $$ all generating the same topology $\mathcal{T}$.

\end{enumerate}
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