Compactness

In a metric space there are usually sequences which do not have cluster points, but there are some metric spaces in which every sequence does have a cluster point. Such spaces are said to be sequentially compact. A related concept is that of compactness and a metric space with this property is called a compact space. As you will see, these two concepts are equivalent for metric spaces, but since they are not equivalent for the more general case of topological spaces, it is customary to study them separately. After you have learned about some of the properties of compact and sequentially compact metric spaces and have proved that they are equivalent, you will apply the results to the metric space R and derive some very important theorems of analysis.