Eritque Arcus Math Notes

compact

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Def pg.346 21.5

A Metric Space (𝑆,𝑑) is compact if each open cover of 𝑆 has finite subcover.

Lemma 21.6

𝑆 is compact ⟹ totally bounded

Thm 21.7

π‘†βŠ‚(𝑋,𝑑) TFAE

  1. 𝑆 compact
  2. 𝑆 is closed, totally bounded
  3. 𝑆 is sequentially compact

Thm 28 OH

For πΆβŠ‚π‘‹ compact and continue function 𝑓:π‘‹βŸΆβ„

  1. supπ‘₯βˆˆπΆπ‘“(π‘₯)<∞
  2. (βˆƒπ‘₯0βˆˆπ‘‹)supπ‘₯βˆˆπΆπ‘“(π‘₯)=𝑓(π‘₯0) according to every continue function reach it’s maximum.

Heine–Borel theorem

π‘†βŠ‚β„π‘‘

  1. 𝑆 is compact, where every open cover has finite subcover
  2. 𝑆 is closed and bounded

Thm

Inverse image of compact set is compact.

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