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{\bf Question}

Discuss Jordan measurability and content of sets of the following
types

\begin{itemize}
\item[i)]
a countable set with one point of accumulation.

\item[ii)]
a countable set with $10^6$ points of accumulation.
\end{itemize}



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{\bf Answer}

Firstly we assume that $S$ is bounded.

\begin{itemize}
\item[i)]
Cover the point of accumulation with a rectangle $R_1$ such that
$\ds |R_1|<\frac{\epsilon}{2}$.  There are a finite number of
points left over.  Cover these by rectangles $R_2, \cdots R_n$
such that $\ds |R_i|<\frac{\epsilon}{2^i}$ then $\ds
\sum_{i=1}^n|R_i|<\epsilon$. So $C^*(S)=0$ and $c_*(S)=0$ also.

\item[ii)]
Cover the $10^6$ points of accumulation by the rectangles $R_1,
\cdots R_{10^6}$ such that $\ds |R_i|<\frac{\epsilon}{2^i}$. There
are a finite number of points left over.  Cover these by
rectangles $\ds R_{10^6}, \cdots R_n$, $\ds
|R_i|<\frac{\epsilon}{2^i}$. Again $\ds
\sum_{i=1}^n|R_i|<\epsilon$. So $C^*(S)=0$ and also $c_*(S)=0$.

\end{itemize}


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