totally bounded

Def 21.1

$S\subset X$ is totally bounded if

$$(\forall \epsilon >0)(\exists m)(\exists x_1,x_2,\dots ,x_m\in X)S\subset \bigcup _{\alpha =1}^m{B}_{\epsilon }(x_{\alpha })$$

where $B_{\epsilon }$ is Delta-ball.

Thm 21.1

totally bounded $⟹$ bounded