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Compactness in Metric space

Compactness in Metric space. Compact. X: metric space. X is compact if and only if any sequence in. X has a subsequence which converges in X. Observation. A compact metric space is complete. Observation. (X, ρ): metric space. is compact if (K, ρ) is compact.

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Compactness in Metric space

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  1. Compactness in Metric space

  2. Compact X: metric space X is compact if and only if any sequence in X has a subsequence which converges in X.

  3. Observation A compact metric space is complete.

  4. Observation (X, ρ): metric space is compact if (K, ρ) is compact. then K is closed and bounded. closed and bounded In Rn, compact see next page But It does not hold in ℓ2

  5. Totally bounded A metric space X is called totally bounded ,there are if for any such that The set is called an ε-net of X.

  6. Observation (X, ρ): metric space boundedness Totally bounded

  7. Theorem If X is a complete metric space, then X is compact if and only if X is totally bounded.

  8. Corollary Let X be a metric space. If X is compact, then X is separable.

  9. Corollary Let X be a compact metric space. The topology of X is generated by a countable number of balls of X.

  10. Theorem A metric space X is compact if and only if every open cover of X contains a finite subcover of X.

  11. C(S) S: compact metric space C(S): the space of all continuous real-valued functions defined on S. For f is uniformly continuous

  12. Uniformly Bounded M is called an uniformly bounded family if M is bounded in C(S)

  13. Equicontinuous M is called equicontinuous if for any ε>0, with d(s,t)<δ there is a δ>0 s.t.

  14. Theorem (Arzelá-Ascoli) is compact A closed set if and only if (i) M is bounded on C(S) (ii) M is equicontinuous

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