By W W. Rouse 1850-1925 Ball

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Inf f{X\ (β) Â ^ s u p / W , (y) λ is adherent to f(X). If (a), then {xeX\f(x) ^ /} - X, hence closed. If (β) or (y), then { . Y G ! Y) ^ / } is empty or equal to f]aef^x) {xeX\ f(x) ^ a} which, as an intersection of closed sets (case 1 ), is itself closed. We may use the same argument for the set {xeX\f(x) ι%λ}. (2) Altering ν to a continuous increasing function. The second part of the proof consists in altering the increasing function ν : X R in such a way that it becomes continuous while remaining increasing.

So far then, we have l the consumption set X(a) is a closed subset of R + , a consumption plan χ is an element of X(a). 2. Preferences Fundamental characteristics of an agent in the sort of economy considered here are his 'tastes' or 'preferences'. All our earlier discussion turned on Ada's and Bill's preferences. When considering two bundles in his consumption set, as we have seen, an agent is able to make one of the following k ς statements : \ prefer χ to y\ I prefer y to x\ or Ι am indifferent between χ and y.

But Ux is the complement of Uy, and hence it is open. The same holds for Uy. This completes the proof. D. The properties of transitivity and continuity just discussed are basic to the whole structure that we will build. We repeat that from now on whenever we refer to a preference relation ^ it will be a complete, transitive, reflexive and continuous binary relation. The set of all such preference relations defined on the positive orthant Rl is denoted by However, there are other conditions on preferences which we will use but which are not, in general, so essential to the theory that we should include them in the actual definition of preferences.