For a set to be closed and bounded in a metric space, it must satisfy two conditions. First, the set must contain all of its limit points, meaning that any sequence of points in the set that converges must also converge to a point within the set. Second, the set must have a finite diameter, meaning that the distance between any two points in the set is bounded by a real number.
Proving that a set is closed and bounded in a metric space is important because it guarantees certain properties about the set. For example, a closed and bounded set will have a well-defined boundary and will not have any gaps or holes. This can be useful in many applications, such as in optimization problems or in proving the existence of certain mathematical objects.
To prove that a set is closed and bounded in a metric space, you must show that it satisfies the two conditions mentioned before: it contains all of its limit points and has a finite diameter. This can be done by using the definitions of closed and bounded sets, as well as the properties of metric spaces, such as the triangle inequality and convergence of sequences.
A set is compact in a metric space if it is both closed and bounded, as well as being "small" in some sense. This means that the set cannot be "spread out" infinitely and must be contained within a finite region of the metric space. Compact sets have many useful properties, such as being sequentially compact and having a finite subcover.
A set can be closed and bounded but not compact in a metric space if it fails to satisfy the "smallness" requirement of compactness. This can happen if the set is too spread out or if it contains "holes" or gaps. An example is the set of all real numbers between 0 and 1, which is closed and bounded but not compact because it does not include the endpoints 0 and 1.