Closure vs. Completeness

So how can we relate these various concepts?

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

We use the standard notation to define the ``Open Balls in s'':

These balls form a Basis for the Metric Topology on s. So, by definition of a limit point, . Thus we can pick a point in this intersection for every , 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 , the set of Rational Numbers. Consider any sequence converging to . The sequence IS Cauchy, and IS the limit of the sequence. BUT 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 , a Closed set is always Complete. Is this always true? By no means.

Example 2   Consider with the Metric Topology, and the following canonically Closed (and Bounded to boot) sets:

Since metrics are real valued, we are entirely within our rights to look at . There are clearly points on the boundary of this set which will be the limit of a convergent sequence in , 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:

Now since Closed sets are sets with Open complement, the set 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, , is simply the set of all infinite sequences x_i of Real numbers with