Limit Points and Completeness

Scot Free Kennedy

Main Idea

Many times one reads that ``this is a topological property, while that is a 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 induce a topology on the space, which in the abstruse way of mathematicians, we generally call the 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).

Basic Theory and Definitions

We begin by recalling a few basic definitions.

Topological Spaces

Definition 1   A Closed Set is a set whose complement is Open.

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.

Definition 2   Given a set a in a space ${\mathcal S}$, and a point $x \in \mathcal S$, x is a Limit Point of $a \subset \mathcal S$ iff every open set containing x has a non-empty intersection with ${\mathcal S}$.

Proposition 1   A set is closed iff it contains all its limit points.

Metric Spaces

Definition 3   A sequence ( x_i ) is a Cauchy Sequence iff

$$\forall \epsilon > 0\ \exists N \in {\mathbb N} \ {\textstyle such \ that} \ \forall i,j > N \ d(x_i,x_j) \leq \epsilon$$

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 can we talk about ``what the sequence is converging to?''

Definition 4   A set s is complete iff every Cauchy Sequence contained in sconverges 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 $o \rightarrow \infty$.

Closure vs. Completeness

So how can we relate these various concepts?

Proposition 2   x₀ a limit point (in the Metric Topology...)of $s \ \Rightarrow \exists$ a Cauchy Sequence contained in s converging to x₀

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.

Example 1   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!

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.

Example 2   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.

OK, that's somewhat contrived. But the further we move from the usual Real topologies, the less contrived our examples become.

Example 3   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.

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.

Definition 5   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

$$\Sigma x_i < \infty$$

Proposition 3  

Connections

Literature

About this document ...

Limit Points and Completeness

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