Def 19.1 A set 𝑆 is not connected if exists open set 𝑂1,𝑂2 such that 𝑆⊆𝑂1∪𝑂2 𝑆∧𝑂1≠∅ 𝑆∧𝑂2≠∅ 𝑆∩𝑂1∩𝑂2=∅ Thm 19.2 𝑆⊂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]⟶ℝ be continue, 𝑦∈[𝑓(𝑎),𝑓(𝑏)]⟹(∃𝑎≤𝑡≤𝑏)𝑝(𝑡)=𝑦