Given a family of sets and a point in the union of the family of sets the neighbourhood of sets around that point is the family of all sets that contain that point as a member. Given the collection of the neighbourhoods of a given family of sets there are two set systems that we can form from the neighbourhoods using union and intersection.
By taking the intersection of each of the neighbourhoods of the family of sets we can get a preorder containment family and by taking the union of each of the neighbourhoods of the family of sets we can get a adjacencies family. Both of these families are nullfree and the preorder containment family is also union free. The adjacencies family is also subunique free as it is subsingleton free in addition to nullfree.
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