2017 year in review

The theory of set systems continued this year. I realized early on in my own mathematical study is that set theory is the best foundation of mathematics, which naturally leads to the theory of set systems. Two types of set systems became of particular interest to me at this point in time: subclass closed set systems and moore families. These two types of set systems seem to play a foundational role, even in the theory of set systems themselves as we can consider subclass closed families of set systems, and moore families of set systems and their properties.

One thing that is worth realizing when dealing with these two types of set systems is that they both have a necessary rank associated with them, which determines the number related elements necessary to define any set in them. The necessary rank of a subclass closed system, is the smallest number of elements needed in a set to determine membership. The necessary rank of Moore family is the smallest number of elements needed to define the closure of a set together.

A power set is both subclass closed and a moore family. It is the smallest necessary rank for both types of set system, and it is trivially defined by its parent set, as it is rank complete, meaning it has no distinguishable elements. Clique families are defined by the cliques of a graph, and Alexandrov families are defined by setting the closure of any singleton set equal to the lower set that contains it in its preorder. Naturally, this demonstrates that graphs and preorders are among the structures that can be represented as set systems in this manner.

Sets of sets like these remain the most important area of study. It is worth mentioning the distance between elements in a graph is determined by the length of the shortest path between them. This naturally leads one to consider notions of distance. Starting in this year, I begun to gain a great interest in the theory of metric spaces and related subjects like geometry, topology, and analysis. My understanding of metric spaces reached only a most basic level, so it was necessary to continue studying it into the next year. This new interest in metric spaces is one of the major takeaways of this year.