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\title{Limit Points and Completeness}
\author{Scot Free Kennedy}
\maketitle

\section{Main Idea}


Many times one reads that ``{\em}this is a {\bfseries topological}
property, while {\em that} is a {\bfseries metric} property.''. What does
this mean?

Well, in general, perhaps one should say: nothing.

That is, consider standard English. We may talk about shape, and we may
talk about color, and we may even use both in describing the same object.
But in general, they are simply different descriptive domains.

However, this isn't entirely satisfactory. For many of the mathematical
structures we may examine, the two concepts are related. For instance, if
we define a metric on a space, that will {\em induce} a topology on the
space, which in the abstruse way of mathematicians, we generally call the
{\em Induced Metric Topology}. 

So how can we get a ``feel'' for the relationship between topological
properties and metric ones? One inroads might be the relationship between
Closed Sets (topological) and Complete Sets (metric).

\section{Basic Theory and Definitions}
We begin by recalling a few basic definitions.

\subsection{Topological Spaces}
\begin{def}
A {\em Closed Set} is a set whose complement is Open.
\end{def}

{\em Remember that a Topology is nothing more or less than a definition of
Open Sets. These may or may not look anything like Open Intervals in the
Real Numbers.}

\begin{def}
Given a set $a$ in a space ${\mathcal S}$, and a point $x \in \mathcal S$,
$x$ is a {\em Limit Point of $a \subset \mathcal S$} iff every open set
containing $x$ has a non-empty intersection with $\mathcal S$.
\end{def}


\begin{prop}
A set is closed iff it contains all its limit points.
\end{prop}

\subsection{Metric Spaces}
\begin{def}
A sequence $( x_i )$ is a {\em Cauchy Sequence} iff
\[ \forall \epsilon > 0\  \exists N \in {\mathbb N} 
\  {\textstyle such \  that} \  \forall i,j > N \   d(x_i,x_j) \leq
\epsilon \]
\end{def}

This is but one of many ways to talk about convergent sequences - but it's
almost always the best. One of it's (many) beautiful subtlties is the
following - nowhere in the definition do we mention the point it converges
to! Our definition is solely the condition that, for any nonzero distance 
$\epsilon$,
the elements of our sequence will eventually all be closer than
$\epsilon$, if we ``go far enough out''.

This is a Big Deal! It also prompts our next definition. When {\em can} we
talk about ``what the sequence is converging to?''

\begin{def}
A set $s$ is {\em complete} iff every Cauchy Sequence contained in $s$
converges to a point in $s$. That is, given  $( x_i )$ a Cauchy Sequence,



\[ 
\{ x_i \} \subset s \Rightarrow
\exists \  x_0 \in s \  {\textstyle such \  that} \  d(x_i,x_0)
\rightarrow 0 \]
as $o \rightarrow \infty$.
\end{def}

\subsection{Closure vs. Completeness}
So how can we relate these various concepts?

\begin{prop}
$x_0$ a limit point (in the Metric Topology...)of $s \ \Rightarrow
\exists$ a Cauchy Sequence contained in s converging to $x_0$
\end{prop}
We use the standard notation to define the ``Open Balls in $s$'':
\[ B_x (\epsilon) := \{y \in s : d(x,y) < \epsilon \} \]
These balls form a Basis for the Metric Topology on $s$. So, by definition
of a limit point, $\forall \epsilon > 0 \ B_x (\epsilon) \cap s \neq
\emptyset$. Thus we can pick a point in this intersection for every
$\epsilon$, yielding the desired Cauchy Sequence.

The VERY IMPORTANT POINT here is that the converse is NOT true: the
existence of a Cauchy Sequence contained in a set $s$ tells us nothing
about the Limit Points of $s$ UNLESS the set is complete.

\begin{example}
Let $s := {\mathbb Q}$, the set of Rational Numbers. Consider any sequence
converging to $\pi$. The sequence IS Cauchy, and $\pi$ IS the limit of the
sequence. BUT $\pi$ is not a Limit Point of $s$ - it's not an element of
$s$, so it can't be! 
\end{example}

Now, what's the relationship between Closed sets and Complete sets? In
$\mathbb R^n$, a Closed set is always Complete. Is this always true? By no
means.

\begin{example}
Consider $\mathbb Q$ with the Metric Topology, and the following
canonically Closed (and Bounded to boot) sets:
\[ \overline{B_x (\epsilon)} :=  \{y \in s : d(x,y) \leq \epsilon \} \]
Since metrics are real valued, we are entirely within our rights to look
at $B_x (\pi)$. There are clearly points on the boundary of this set
which will be the limit of a convergent sequence in $B_x (\pi)$, but NOT a
limit point.
\end{example}
OK, that's somewhat contrived. But the further we move from the usual Real
topologies, the less contrived our examples become.

\begin{example}
Now consider the Real line, but with the ``Lower Limit'' Topology. This is
the topology generated by the ``half open intervals'', ie, by the
following basis:
\[ [a,b) := {x \in \mathbb R : a \leq x < b} \ a,b \in \mathbb R \]
Now since Closed sets are sets with Open complement, the set
$ s := (- \infty ,a) \cup [b,\infty)$ is Closed (this is tricky - note
that $a$ is NOT a Limit Point of $s$. Why?). But it is clearly not
complete - take the sequence
\[x_i := a - 1/i \] for example.
 \end{example}
Still a little contrived, but less so. The Lower Limit Topology isn't all
that wierd, really. But now, the best has been saved for last, and this is
really the kind of example that makes these subtle distinctions so
important. 

\begin{def}
The space of Square-Summable Real Sequences, $\ell^2 (\mathbb R)$, is
simply the set of all infinite sequences $x_i$ of Real numbers with 
\[ \Sigma x_i < \infty \]


\end{def}




\begin{prop}


\end{prop}


\section{Connections}


\section{Literature}




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