Nested Sets

This is a nearly trivial piece of set theory. A nested set N is a set of sets such that:

a∊N ∧ b∊N → a⊂b ∨ b⊂a ∨ a∩b = ∅ Some trivial theorems are:

(N is nested) → ({N}∪N is nested)
(N is nested) ∧ M⊂N → (M is nested)

The smallest set of sets of integers that is not nested is {{0, 1}, {1, 2}}

There is a relational data base notion of nested set model which is a programming pattern, suitable for relational data base systems, for naming members of finite nested sets in our sense here.

Our notion imposes no finite limitations on the number of children of a set or even the number, or order type of generations.

A non trivial theorem is that every base for the topology of p-adic numbers has a subset which is also a base for that topology and is also nested. This is not so for the topology the real line.