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shared by the members of T: i ( p ) G i ( T ) = U { i ( x )I X E T } . Show that this prescription satisfies the axioms of betweenness. This leads to a convexity consisting of all sets C E S which are stable in the sense that p E C whenever p has social affinity with a finite set of persons in C. Conversely, show that every convex structure arises from an “interest” function as described above. Hint: be interested in concave sets.
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Further Topics 2.20. Convexity and social affinity (Aumann ). Let S (for “society”) and I (for “interests”) be sets, and let i : S -+ 2’ be a function such that i ( x ) # 0 for all x E S. A person p has social finit)‘ with a finite set of individuals T E S provided his personal interests i ( p ) are
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Chapter 1: Abstract Convex
Structures, p44, pdf 60 of 556. Copied from scanned pdf and pasted below - sorry I don't have time to format properly but I am not competent to work on writing it up for publication and anyone potentially interested in doing so can easily follow the links provided above.
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The "Definition via convergent filters" yields pretopological spaces, which - as their name indicates - are not (yet) topological spaces. Topological spaces may be defined by way of convergence of filters, but this needs a more involved axiom. The relation with topological spaces mentioned in this
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My impression is that the definition below first defines closed sets and extends it to abstract convexity (which is just an intersection closure space plus upwards chain closure). But the original idea by Aumann was for teaching definitions in topology.
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3 Contact
Relations If we formulate Aumann’s original definition of a contact relation given in in our notation, then a relation A : X ↔ 2X is an (Aumann) contact relation if the following conditions hold.
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The page on
Kuratowski closure operators, mentions interior, exterior, and boundary operators, but only gives axioms for the interior operators, which are also on this page (
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3. Each point is in contact with any subset that all the members of any subset it is in contact with are all in contact with.
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and was intended as a more intuitive definition for learning about closed sets as sets that a point is in "contact" with.
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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https://en.wikipedia.org/Kuratowski_closure_axioms#Interior,_exterior_and_boundary_operators
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I would summarize as a relation A from points to subsets is a "contact relation" if:
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2. Each point is in contact with any superset of any subset it is in contact with.
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Contact
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1. Each point is in contact with the subset consisting of itself.
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Add exterior, boundary, derived set, and co-derived set operators.
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