Compactness in Metric Spaces

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Compact Sets in a Metric Space

If is a metric space and then a cover or covering of is a collection of subsets in such that:

Furthermore, we said that an open cover (or open covering) is simply a cover that contains only open sets.

We also said that a subset is a subcover/subcovering (or open subcover/subcovering if is an open covering) if is also a cover of , that is:

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We can now define the concept of a compact set using the definitions above.

Definition: Let be a metric space. The subset is said to be Compact if every open covering of has a finite subcovering of .

In general, it may be more difficult to show that a subset of a metric space is compact than to show a subset of a metric space is not compact. So, let's look at an example of a subset of a metric space that is not compact.

Consider the metric space where is the Euclidean metric and consider the set . We claim that this set is not compact. To show that is not compact, we need to find an open covering of that does not have a finite subcovering. Consider the following open covering:

Clearly is an infinite subcovering of and furthermore:

Let be a finite subset of containing elements. Then:

Let . Then due to the nesting of the open covering , we see that:

But for we need . But , so and , so . Therefore any finite subset of cannot cover . Hence, is not compact.