path connected

Def Path connected 19.5

$S\subset X$ is path connected if $x\ne y\in S$ $(\exists \gamma :[0,1]⟶S)\gamma$ continue and $\gamma (0)=x$ and $\gamma (1)=y$

Thm 19.6

path connected $⟹$ connected

Example

we can prove it’s path connected by manually making a path. e.g.
$f$ as upper bound and $g$ as lower bound, we can make 3 paths and concat them.

1. go up to $f$ from start $x$ coordination
2. go through $f$ to end $x$ coordination
3. go down to end $y$ coordination