Unformatted text preview: % we have z % x and y % z and thus, by transitivity, y % x , contradicting the assumption that x Â y . Comment on the Proof: Another proof could be given for the more general case, in which the assumption that the set X is convex is replaced by the assumption that it is a connected subset of < n . Remember that a connected set cannot be covered by two disjoint open sets. If there is no z such that x Â z Â y , then X is the union of two disjoint sets { a  a Â y } and { a  x Â a } , which are open by the continuity of the preference relation. Recall that a set Y ⊆ X is dense in X if in every open subset of X there is an element in Y . For example, the set Y = { x ∈ < n  x k is a rational number for k = 1, .. , n } is a countable dense set in < n ....
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 Fall '10
 aswa
 Topology, Utility, Metric space, Topological space, Debreu

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