SUMMARY
A metric space consisting of two points, X={a,∞}, is not totally bounded due to the infinite distance d(a,∞)=∞. The discussion highlights that while totally bounded implies bounded, the assumption of finite distances is crucial in this context. The proof referenced from UCLA's resources assumes finite distances between points, which is a fundamental aspect of metric space definitions. Therefore, the metric space X={a,∞} contradicts the property of total boundedness.
PREREQUISITES
- Understanding of metric spaces and their definitions
- Familiarity with the concept of boundedness in mathematics
- Knowledge of the implications of totally bounded spaces
- Basic comprehension of mathematical proofs and axioms
NEXT STEPS
- Study the properties of metric spaces in detail
- Learn about the implications of boundedness and total boundedness
- Review the axioms of metric spaces and their significance
- Examine examples of totally bounded spaces and their characteristics
USEFUL FOR
Mathematicians, students of topology, and anyone interested in the properties of metric spaces and their implications in mathematical analysis.