How do closed, bounded, and compact concepts relate in metric spaces?

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SUMMARY

The discussion clarifies the relationships between closed, bounded, and compact sets in metric spaces, specifically in R^n. A set can be bounded but not closed, exemplified by the interval (0, 1), while the set [0, ∞) is closed but neither bounded nor compact. In R^n, a set is compact if and only if it is closed and bounded. Furthermore, in metric spaces, compactness is equivalent to being complete and totally bounded, where a totally bounded set ensures every sequence has a Cauchy subsequence.

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  • Understanding of metric spaces
  • Familiarity with concepts of closed and bounded sets
  • Knowledge of compactness in topology
  • Basic principles of sequences and convergence
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  • Explore the relationship between completeness and compactness in R^n
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Mathematicians, students of topology, and anyone interested in the foundational concepts of metric spaces and their properties.

Ratzinger
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Could someone explain me how these three concepts hang together?

(When is a set bounded but not closed, closed but not bounded, closed but not compact and so one?)
 
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Examples (real line with usual topology).

Bounded not closed: 0<x<1
Closed not bounded or compact: 0<=x<oo.
 
in R^n, compact is equivalent to closed and bounded, so a closed set is bounded iff compact, and a bounded set is closed iff compact.

in a metric space, compact is equivalent to complete and totally bounded.

in R^n which is itself complete, closed is equivalent to complete, and since every bounded set in R^n has comoact closure, bounded is equivalent to totally bounded.

a totally bolunded set is one in which everys equence ahs a cauchy subsequence, and a completes et one in which everyu cauchy sequence converges.

the connection is that a compact metric space is one in which every sequence has a convergent subsequence. (i think. it has been a long time since i taught this course.)
 

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