Understanding Compactness in Metric Spaces: Closed, Bounded, and Open Covers

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SUMMARY

The discussion centers on the concept of compactness in metric spaces, specifically addressing the relationship between closed and bounded sets and compactness. While it is true that in R^n, closed and bounded sets are compact, this does not hold in all metric spaces. The example provided involves the subspace Y = (0,1) of X = R, where A = (0,1/2] is closed and bounded in Y. The confusion arises regarding the choice of open covers, whether they should be relative to X or Y, highlighting the importance of definitions in metric space theory.

PREREQUISITES
  • Understanding of metric spaces and their properties
  • Familiarity with the concepts of closed and bounded sets
  • Knowledge of open covers and their role in compactness
  • Basic comprehension of Euclidean metrics
NEXT STEPS
  • Study the definition and properties of compactness in various metric spaces
  • Explore examples of non-compact sets in different metric spaces
  • Learn about the Heine-Borel theorem and its implications for compactness
  • Investigate the role of open covers in determining compactness
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Mathematicians, students of topology, and anyone interested in the properties of metric spaces and compactness will benefit from this discussion.

Buri
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My professor in lecture yesterday said that if a set is closed and bounded in a metric space it doesn't necessarily imply that it is compact. If X = R^n, then it does happen to be true, however. I was trying to construct an example, but I am getting confused. If I let X = R, and Y = (0,1) where Y is a subspace of X, then A = (0,1/2] is closed and bounded in Y. However, from where do I choose the open cover? That is, open relative to X or Y? I know in this case it won't make a difference, but maybe in differently chosen X, Y and A it might. I guess this is a matter of definition, but would like some help.

Thanks a lot.
 
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Don't think about a subset of Rn. Metric spaces just are, they aren't normally viewed as being in a larger metric space even if that is useful. In this case you have X=(0,1) with the Euclidean metric, and A=(0,1/2]. The fact that you can embed this metric space into the real numbers is irrelevant
 
Ahh I see. Thanks!
 

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