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I have seen many different set systems: sigma algebras, topologies, closure systems, $\pi$-systems, monotone classes, fields of sets, bornologies, independence systems, feasible systems, matroids, greedoids, convexity structures, .... I bet there are even more. So I think an organization of them will be of great help.

I was wondering if there are some references that explain the relations or connections between them or most of them, and their classifications, preferably at the level of set systems?

The references can come from

  • a book entirely devoted to general set systems, or
  • a chapter regarding general set systems in a book in some specific area, or
  • a paper survey, or
  • even a diagram pointing out their relations will be nice. Just like some diagram relating different probability distributions, some diagrams relating different mathematics branches, ...

Thanks and regards!

Tim
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  • Maybe you could start by grouping your set systems: $\sigma$-algebras, $\pi$-systems, monotone classes and fields of sets belong to measure-theory/probability. Bornologies and convexity structures to functional analysis (and metric geometry?). The rest seems to be combinatorics. So: get books on these topics. Can you be more explicit about what kind of relations you are looking for? – Martin Jan 03 '13 at 19:43
  • @Martin: Thanks! (1) The areas you point out some set systems belong to are their applications, I think. There may even be interactions between set systems applied to different areas, I suspect. Or is a set system just used for one purpose/application? (2) Examples of relations between set systems would be: what set system is defined in terms of what set system, such as convexity structure is a special kind of closure system; in what conditions one set system becomes/generates another, such as the relation between sigma algebras, pi systems and monotone classes. – Tim Jan 03 '13 at 19:51
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    Yes, this is exactly the kind of grouping I mean. This is already more specific than a loose and more or less random list of special collections of subsets of a given set... Next step: think about some examples why you even care about these things. Applications are not bad! Concrete instances lead to an understanding of the abstraction. – Martin Jan 03 '13 at 19:59
  • Why I even care about these things is to understand why there exist so many different set systems, and so few sources I can find to summarize/survey them in a nutshell. Yes, applications and examples are not bad. They are part of reasons for so many set systems. – Tim Jan 03 '13 at 20:04
  • I have thought about these "set systems" from time to time (but not in any serious way). What one can do is to put all of these structures into a category. The category would be pairs, $(S,\tau)$, where $\tau\subset\mathcal{P}(S)$. The morphisms, $(S,\tau)\rightarrow (S',\tau')$ are set maps, $f:S\rightarrow S'$ such that for every $X\in\tau'$, $f^{-1}(X)\in\tau$. Sitting inside of this category is topological spaces, Borel algebras, filters and ultrafilters and some that you have mentioned (if not all) as full subcategories. Furthermore many – Baby Dragon Jan 04 '13 at 04:16
  • (Continued)of these have adjoints (I'm a bit dyslectic about left or right at the moment, I think right adjoints). So learning about some category theory would be helpful when you map out your diagram. – Baby Dragon Jan 04 '13 at 04:16
  • @BabyDragon: Thanks for providing this unifying point of view! – Tim Jan 04 '13 at 05:05

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