Equivalent statements to compactness of a metric space

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Theorem statement

Given a metric space (X,d), the following are equivalent[1][Note 1]:

  1. X is compact
  2. Every sequence in X has a subsequence that converges (AKA: having a convergent subsequence)
  3. X is totally bounded and complete

Proof

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1)2): X is compact (an)n=1X  a sub-sequence (akn)n=1 that coverges in X

[Expand]

2)3): Suppose for all sequences (xn)n=1X that (xn)n=1 has a convergent subsequence (X,d) is a complete metric space and is totally bounded



TODO: Rest, namely: 31



Notes

  1. Jump up To say statements are equivalent means we have one one of the other(s)

References

  1. Jump up Introduction to Topology - Theodore W. Gamelin & Robert Everist Greene