connected

Def 19.1

A set $S$ is not connected if exists open set $O_1,O_2$ such that

1. $S\subseteq O_1\cup O_2$
2. $S\wedge O_1\ne \varnothing$
3. $S\wedge O_2\ne \varnothing$
4. $S\cap O_1\cap O_2=\varnothing$

Thm 19.2

$S\subset \mathbb{R}$ is connected $⟺$ $S$ is interval

Thm 19.3

for $f:X⟶Y$ continue, $S$ connected $⟹$ $f(S)$ connected

Thm Intermediate value 19.4

$f:[a,b]⟶\mathbb{R}$ be continue, $y\in [f(a),f(b)]⟹(\exists a\le t\le b)p(t)=y$