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\centerline{Additions to Part IV: Notes}
\bigskip
Add the following as a new paragraph at the end of note 40:
\smallskip
The following are definitions for [94 S]. Suppose $x\in\Bbb R$
and $X\subseteq \Bbb R$.
\itemitem{(a)} $x$ is called an {\it accumulation point} of $X$ if
every open neighborhood of $x$ contains a point in $X-\{x\}$.
\itemitem{(b)} $x$ is called a {\it cluster point} of $X$ if every
open neighborhood of $x$ contains an infinite number of points in
$X$.
\itemitem{(c)} $x$ is called a {\it limit point} of $X$ if there
is a sequence $\{x_n: n\in\omega\}\subseteq X$ such that for every
open neighborhood $N_x$ of $x$, there is real number $M > 0$ such
that for all $n > M$, $x_n\in N_x$.
\smallskip
In any T$_1$ space, ($\Bbb R$ with the order topology is T$_1$),
a point is a cluster point if and only if it is an accumulation
point. (Clearly, a cluster point is an accumulation point.
Suppose $x$ is an accumulation point of $X$ that is not a cluster
point. Suppose $N_x$ is a neighborhood of $x$ that only contains
a finite number of elements of $X$, $x_1$, $x_2$, $\ldots$, $x_n$.
Since $X$ is T$_1$, using induction, we can find a neighborhood $M$
of $x$ that does not intersect $\{x_1, x_2,\ldots, x_n\}$. Then
$N_x\cap M$ is a neighborhood of $x$ that does not contain any
points of $X$ different from $x$. This contradicts the fact that
$x$ is an accumulation point.)
\bye