Example of a Non-Empty Open and Closed Set in Metric and Normed Spaces

In the realm of mathematical analysis, particularly within the context of metric spaces and normed spaces, the concepts of open and closed sets are fundamental. A set is considered both open and closed (clopen) if it satisfies the criteria for both properties within the given space.

Understanding Open and Closed Sets

To clarify, a set ( A ) in a metric space ( X ) is open if every point in ( A ) has a neighborhood entirely contained in ( A ). Formally, for each point ( x in A ), there exists an ( epsilon > 0 ) such that the open ball ( B(x, epsilon) ) centered at ( x ) with radius ( epsilon ) is a subset of ( A ). On the other hand, a set ( A ) is closed if its complement in the space ( X ) is open. This implies that ( A ) contains all its limit points.

Special Cases: The Empty Set and the Whole Space

It is a well-known fact that the only sets that are both open and closed in a given metric space or normed space are the empty set ( emptyset ) and the entire space ( X ). The empty set is trivially both open and closed because its complement is the entire space, and any subset of an empty set is both open and closed. However, the empty set is not considered non-empty by definition.

Non-Empty Clopen Sets in Metric and Normed Spaces

The only non-empty set that is both open and closed in a metric or normed space is the entire space ( X ) itself. This can be understood by considering the properties of open and closed sets. If a set ( A ) is open and closed, its complement ( X setminus A ) must also be both open and closed. Therefore, ( A ) and its complement must partition the space such that the only way this can happen is if ( A ) is either the entire space or the empty set.

Proving the Uniqueness of Non-Empty Clopen Sets

To formally prove this, consider a non-empty set ( A ) in a metric space ( X ) which is both open and closed. Since ( A ) is closed, its complement ( X setminus A ) is open. Now, ( A ) and ( X setminus A ) are disjoint and their union is ( X ). If both ( A ) and ( X setminus A ) are non-empty, this would imply that ( X ) can be partitioned into two non-empty open sets, which is a contradiction. Therefore, if ( A ) is non-empty and both open and closed, ( A ) must be the entire space ( X ).

Applications in Topology

The concept of clopen sets is crucial in various areas of mathematics, including topology. In topology, clopen sets are significant in the study of connectedness and the construction of topological spaces. A space is called totally disconnected if the only connected subsets are the singletons and the whole space itself, where the only clopen sets are the empty set and the space.

Conclusion

Therefore, the only non-empty set that is both open and closed in a metric or normed space is the entire space itself. This result is a fundamental property in the study of these spaces and has important implications in various areas of mathematics. Understanding this property is crucial for grasping more advanced concepts in topology and analysis.