complete

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 $\underline{Y}$ and $\varphi :X⟶\underline{Y}$

1. d_\underline{Y}(\phi(x), \phi(y)) = d(x, y)
2. $\varphi (x)$ is dense in $\underline{Y}$

Def 18.3

completion is unique(too complex to prove)