Meanings of root system: Show definitions/intuitions and $Λ_{root} ⊂ Λ_{char} ⊂ Λ_{weight}?$

Question

Let G be a compact Lie group. Let $g_{\mathbb{C}}$ be the associated complex Lie algebra. There is a root system associated to $g_{\mathbb{C}}$. Denote this by a 4-tuple $$R = \{Λ_{root}, Λ_{weight}, Λ_{co−root}, Λ_{co−weight}\}.$$

$T$ is a maximal torus of G,

$Λ_{char }= Hom(T,U(1))$.

$Λ_{cochar }= Hom(U(1),T)$.

Can you help on showing/explaining:

1. Definitions and intuitions of $Λ_{root}, Λ_{char}, Λ_{weight}$?

2. Definitions and intuitions of $Λ_{coroot}, Λ_{cochar}, Λ_{coweight}$?

3. How to show in general, $$Λ_{root} ⊂ Λ_{char} ⊂ Λ_{weight}?$$

4. How to show in general, $$Λ_{coroot} ⊂ Λ_{cochar} ⊂ Λ_{coweight}?$$

5. When would some of $Λ_{root}, Λ_{char}, Λ_{weight}$ and some of $Λ_{coroot}, Λ_{cochar}, Λ_{coweight}$ overlap to be the same?