closure

Def 17.1

$\bar{S}=\{{\lim }_k{x}_k\mid x_k\in S\}\cup S$
is closure of set $S$, it has following properties

1. $\bar{S}$ is closed
2. $\bar{S}$ is smallest closed set which contain set $S$

Def 18.1

$S$ is dense in $\bar{S}$
$(\forall y\in \bar{S})(\exists x\in S)(\forall \epsilon >0)d(x,y)<\epsilon$