Def Path connected 19.5

𝑆𝑋 is path connected if 𝑥𝑦𝑆 (𝛾:[0,1]𝑆)𝛾 continue and 𝛾(0)=𝑥 and 𝛾(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