compact

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

$S\subset (X,d)$ TFAE

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

Thm 28 OH

For $C\subset X$ compact and continue function $f:X⟶\mathbb{R}$

1. ${\sup }_{x\in C}f(x)<\infty$
2. $(\exists x_0\in X){\sup }_{x\in C}f(x)=f(x_0)$ according to every continue function reach it’s maximum.

Heine–Borel theorem

$S\subset {\mathbb{R}}^d$

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.