A Brunnian braid is a braid that becomes trivial upon removal of any one of its strings. My question:
Are there any other example in other fields of mathematics that carries the similar idea?
I was thinking about some mathematical object which consists of some (possibly infinite) components (generators, etc.) such that deleting any one of them results in a trivial object, just as trivial braids (or trivial links). For example, a group with $n$ generators $x_1,\ldots,x_n$which becomes trivial (or a cyclic group) after quotient out the normal subgroup generated by any $x_i$.