Def 14.2

A metric space is complete if every Cauchy sequence in 𝑋 is convergent to some points in the set 𝑋.

Thm 15.1

For 𝑋(𝐶,𝑑) a Metric Space
TFAE

  1. 𝐶 is closed
  2. (𝐶,𝑑) is complete

Thm 17.6 Existence of completion

For (𝑋,𝑑) is a Metric Space, then there exists a complete Metric Space 𝑌 and 𝜙:𝑋𝑌

  1. 𝑑𝑌(𝜙(𝑥),𝜙(𝑦))=𝑑(𝑥,𝑦)
  2. 𝜙(𝑥) is dense in 𝑌

Def 18.3

completion is unique(too complex to prove)