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%\caption{A pox on both your houses!}
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\framebox{
\raisebox{0in}[.5in][.2in]{
\textbf{MA 314 Homework 15: Bad Quotient! Bad! Bad!} 
}
}



\section{Reading}

\begin{enumerate}\spacing
\item Read some sources which you'll use as the basis for your blog essay.
\end{enumerate}

\section{Bricks}

\begin{enumerate}\spacing
\item In class we considered two bad quotients:
\begin{itemize}\spacing
\item $X = \R$ with the equivalence relation: $x \sim y \Leftrightarrow (x-y) \in \Q$. This has the property that in $\ob{X}$ there are distinct points $\ob{P}$ and $\ob{Q}$ such that the chain $\ob{C}: \ob{P}, \ob{Q}$ of jump number 1 has length 0.

\item $X = [0,1] \times [-1,1] \subset \R^2$ with the equivalence relation generated by $(1/n,1) \sim (1/n, -1)$. This has the property that in $\ob{X}$ there are distinct points $\ob{P}$ and $\ob{Q}$ such that for any $\epsilon > 0$, there is a chain $\ob{C}: \ob{P}, \ob{x}, \ob{Q}$ of jump number 2 with $\ell(\ob{C}) < \epsilon$. But there there exists an $\epsilon > 0$ such that no chain of jump number less than 2 (i.e. jump number 1) has length less than $\epsilon$.
\end{itemize}

Construct a space $X$ and an equivalence relation giving rise to a quotient space $\ob{X}$ such that both of the following hold:
\begin{enumerate}\spacing
\item There exist distinct points $\ob{P}, \ob{Q} \in \ob{X}$ such that for any $\epsilon > 0$, there is a chain $\ob{C}$ from $\ob{P}$ to $\ob{Q}$ of jump number $j = 3$ such that $\ell(\ob{C}) < \epsilon$.
\item For all pairs of distinct points $\ob{P}, \ob{Q} \in \ob{X}$ there exists $\epsilon > 0$ such that if $\ob{C}$ is a chain from $\ob{P}$ to $\ob{Q}$ of jump number $j(\ob{C}) < j$ then $\ell(\ob{C}) > \epsilon$.  
\end{enumerate}




\end{enumerate}
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