An infinite profinite group is just infinite if every nontrivial closed normal subgroup has finite index, equivalently is open. Requiring closedness is part of the topological convention. This property says that every proper Hausdorff quotient is finite.
An infinite profinite group is hereditarily just infinite if every open subgroup is a just-infinite profinite group. To prove this, it suffices to show that every nonidentity element of every open subgroup has open closed normal closure: any nontrivial closed normal subgroup then contains that normal closure. The Nottingham group is an example. In odd characteristic its depth-commutator formula constructs nonzero elements of each sufficiently high layer inside any such normal subgroup, and successive coefficient correction fills the entire tail.
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