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\title{Problem Set 2: Topological spaces}
\date{Due: Thursday, February 11}
\author{Your name:}





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\maketitle



\begin{prob}[7]
Let $(M,d)$ be a metric space, and let $x$ be a point in $M$. Show that the subset $M \setminus \{ x \}$ is open in the metric topology $\t_d$.
\end{prob}

\begin{prob}[12]
Let $X$ be a space.
\begin{enumerate}
	\item Suppose $(\t_i)_{i \in I}$ is a family of topologies on $X$ indexed by $I$. Prove that $\bigcap_{i \in I} \t_i$ is a topology on $X$.
	\item Suppose $\t, \t'$ are topologies on $X$. Is $\t \cup \t'$ a topology on $X$? Justify your claim.
	\item Let $\mathcal{A}$ be a basis for a topology on $X$, and let $I$ be the collection of topologies $\t$ on $X$ such that $\mathcal{A} \subset \t$. Prove that $\t_{\mathcal{A}} = \bigcap_{\t \in I} \t$. In other words, $\t_A$ is the \emph{coarsest} topology that contains $\mathcal{A}$. Is this true if $\mathcal{A}$ is only a sub-basis for a topology on $X$?
\end{enumerate}
\end{prob}


\begin{prob}[18]
A totally ordered set is a set $X$ together with a subset $R \subset X \times X$ satisfying the following properties:
\begin{itemize}
	\item for every $x \in X,$ $(x,x)$ does \emph{not} belong to $R$;
	\item for every $x,y \in X$, exactly one of $(x,y)$ and $(y,x)$ belongs to $R$.
	\item for every $x,y,z \in X$, if $(x,y)$ and $(y,z)$ belong to $R$ then $(x,z)$ belongs to $R$. 
\end{itemize}
We often write $x < y$ as an abbreviation for $(x,y) \in R$. The subset $R$ is called the \emph{total ordering}. Given $a,b \in X$, we define the following subsets of $X$:
\begin{align*}
	(-\infty, a) & \equiv \{ x \in X : x < a \} \\
	(a, \infty) & \equiv \{ x \in X : a < x \} \\
	(a,b) & \equiv (a, \infty) \cap (-\infty, b).
\end{align*}
Denote by $\mathcal{B}$ the collection of all such subsets, as well as $X$ itself.
\begin{enumerate}
	\item Show that $\mathcal{B}$ is a basis for a topology $\t_{R}$ on $X$.
	\item Recall that $\R$ has a total ordering, $x < y \iff y - x$ is positive. Prove that the order topology on $\R$ coincides with the standard (metric) topology on $\R$.
	\item Note that $\R^2 = \R \times \R$ has a total order defined as follows: $$ (a,b) < (c,d) \iff a < c \text { or } (a = c \text{ and } b <d) .$$ This is called the \emph{lexicographical order} (think of how words are ordered in the dictionary). Denote this ordering $L \subset \R^2 \times \R^2$. Observe that the subset $C = [0,1] \times [0,1] \subset \R^2$ inherits a total ordering from $\R^2$, which we denote by $T \subset C \times C$. This means we have \emph{three} topologies on $[0,1] \times [0,1]$: 
		\begin{itemize} 
			\item $(\t_{L})_{[0,1] \times [0,1]}$, the subspace topology from the ordering on $\R^2$;
			\item $\t_T$, the topology from the induced ordering on $[0,1] \times [0,1]$;
			\item the subspace topology on $[0,1] \times [0,1]$ from the standard topology on $\R^2$. 
		\end{itemize}
Compute the closure of the set $A = \{ (x,0) : x \in [0,1) \}$ in each of these topologies.
\end{enumerate}
\end{prob}

\begin{prob}[18]
Let $X$ and $Y$ be topological spaces. 
\begin{enumerate}
	\item Let $\mathcal{B}_{\prod}$ be the collection of subsets of $X \times Y$ of the form $U \times V$,where $U$ is open in $X$ and $V$ is open in $Y$. Show that $\mathcal{B}_{\prod}$ is a basis for a topology on $X \times Y$. This topology is called the \emph{product topology}.
	\item  Let $\mathcal{B}_{\coprod}$ be the collection of subsets of $X \coprod Y$ of the form $U \times \{0\}$ or $V \times \{1\}$,where $U$ is open in $X$ and $V$ is open in $Y$. Show that $\mathcal{B}_{\coprod}$ is a basis for a topology on $X \coprod Y$. This topology is called the \emph{sum topology}.
	\item Consider $\R$ equipped with its standard metric topology. Show that the product topology on $\R \times \R$ is the same as the standard metric topology on $\R^2$.
\end{enumerate}
\end{prob}

\begin{prob}[20]
The goal of this problem is to show that there are infinitely many prime numbers. This result is known as {\color{plum} Euclid's theorem}, its first recorded proof having been published by Euclid around 300 B.C. The surprising topological proof that we will see here was discovered in 1955 by H. Furstenberg.

Recall that a natural number $p \in \N$ is \emph{prime} iff $p \neq 1$ and if its only divisors are $1$ and $p$. Let $\mathbb{P} \subset \N$ be the set of all prime numbers.

\begin{enumerate}
	\item Show that every natural number $n \neq 1$ is divisible by a prime number.
	\item For $x, n \in \Z$ let $$x + n\Z = \{x + nz : z \in \Z \}.$$ Call a subset $U \subset \Z$ open iff for every $x \in U$ there exists $n \in \N \setminus \{0\}$ such that $x + n\Z \subset U$. Show that this defines a topology on $\Z$.
	\item Show that for every $n \in \N \setminus \{0\}$, the set $n\Z$ is both open and closed in $\Z$.
	\item Using (a), show that $$\Z \setminus \{1,-1\} = \bigcup_{p \in \mathbb{P}} p\Z .$$
	\item Conclude that $\mathbb{P}$ is infinite.
\end{enumerate}

\end{prob}




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