Equivalent statements to compactness of a metric space
From Maths
Contents
[hide]Theorem statement
Given a metric space (X,d), the following are equivalent[1][Note 1]:
- X is compact
- Every sequence in X has a subsequence that converges (AKA: having a convergent subsequence)
- X is totally bounded and complete
Proof
[Expand]
1)⟹2): X is compact ⟹ ∀(an)∞n=1⊆X ∃ a sub-sequence (akn)∞n=1 that coverges in X
[Expand]
2)⟹3): Suppose for all sequences (xn)∞n=1⊆X that (xn)∞n=1 has a convergent subsequence ⟹ (X,d) is a complete metric space and is totally bounded
TODO: Rest, namely: 3⟹1
Notes
- Jump up ↑ To say statements are equivalent means we have one ⟺ one of the other(s)
References