SUMMARY
The discussion focuses on proving that the set of integers with the discrete metric \( p(m,n) = 1 \) (for \( m \neq n \)) and \( p(n,n) = 0 \) is closed and bounded but not compact. The set is closed because it contains all its limit points, which are nonexistent in this case. It is bounded as one can find a ball that encompasses all integers. However, it is not compact since there exists an open cover that lacks a finite subcover, specifically by using open balls of radius greater than one centered at any integer.
PREREQUISITES
- Understanding of metric spaces and their properties
- Familiarity with the concept of compactness in topology
- Knowledge of closed sets and limit points
- Basic principles of open covers and finite subcovers
NEXT STEPS
- Study the properties of discrete metric spaces
- Learn about compactness criteria in different types of metric spaces
- Explore examples of closed and bounded sets in various metrics
- Investigate the implications of the Heine-Borel theorem in Euclidean spaces
USEFUL FOR
Mathematics students, particularly those studying topology and metric spaces, as well as educators looking for examples of closed and bounded sets that are not compact.