Separable Metric Spaces

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Dense Sets in a Metric Space

We will now look at a new concept regarding metric spaces known as dense sets which we define below.

Definition: Let be a metric space and let . Then is said to be Dense in if for every and for every we have that , i.e., every open ball in contains a point of .

In any metric space the whole set is always dense in . Furthermore, the empty set is not dense in .

For a less trivial example, consider the metric space where is the usual Euclidean metric defined for all by , and consider the subset of rational numbers.

The set is dense in because for any open ball, i.e., for any and for any we have that the open interval contains a rational number.

For a counterexample, consider the set of integers. We claim that is not dense in . To show this, consider the following ball:

Clearly and so is not dense in .

We will now look at a nice theorem which tells us that for a metric space a set is dense in if and only if its closure equals .

Theorem 1: Let be a metric space and let . Then, is dense in if and only if .

Recall that denotes the closure of , and we defined the closure of to be the set of adherent points of .

  • Proof: Suppose that is dense in . Then for all and all we have that:
  • So every is an adherent point of . The set of all adherent points of is the closure of , so .
  • Suppose that . Then every point of is an adherent point of , i.e., for all and for all we have that:
  • Therefore is dense in .

Separable Metric Spaces

Recall that if is a metric space then a subset is said to be dense in if for every and for all we have that:

In other words, is dense in if every open ball contains a point of .

We will now look at a special type of metric space known as a separable metric space which we define below.

Definition: A metric space is said to be Separable if there exists a countable dense subset of .

For example, consider the metric space where is the usual Euclidean metric defined for all by . Then the subset is dense in since every open interval contains rational numbers.

In fact, in general, the metric space where is the usual Euclidean metric defined for all by:

Then it can be shown similarly that the following set is dense in :


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