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absorbing set, 230 closed correspondence, 134 abstract economy, 320 closed function, 141 as coarse as (topologies), 4 Closed Graph Theorem, 214 as fine as (topologies), 4 closed sections, 175 as fine as (bases), 15 closed sets, 7 Axiom of Choice, 188 closure (of a set), 8 Baire Category Theorem, 37 competitive equilibrium, 256 balanced set, 230 complete metric space, 34 Banach Space, 190 composition, correspondences, 150 conjugate index, 185 Banach-Steinhaus Theorem, 205 connected space, 94 base at a point, 231 continuous correspondence, 119 basis (for a topology), 14 convergence: basis generated by ti, 232 in metric space, 34 Berge's Maximum Theorem, 280 sequences in top. spaces, 43 best-reply correspondence, 318 of a net, 44 bounded linear function, 198 correspondence, 111 bounded subset, 87 closed-valued, 118 box topology, 59 compact-valued, 118 Brouwer-Schauder-Tychonoff nonempty-valued, 118 Theorem, 305 open-valued, 118 chain, 189 countable set, 28 closed sphere, 35 covering, 61 cluster point, dense, 28 of a set, 10 dense cone, the, 251 of a sequence 43 denumerable, 28 of a net, 44 diameter of a set, 91 collective response corresp., 319 directed system, 43 compact, 61 distance function, 32 continuous, dual space, 206 binary relation, 24 function (topological spaces), 18 (Nash) equilibrium for a game, 318 function at a point, 19 for an abstract economy, 320 selection, 306 equivalent bases, 15 contraction mapping, 38 extension of a function, 207 countably compact, 73 face (of a simplex), 288 338 INDEX feasible allocation, 256 lower inverse, 112 finite intersection property, 62 maximal element, 189 fixed point, 302 metric, of a contraction mapping, 38 definition, 25 functional, 206 space, 25 gauge of a set, 248 topology, 26 generalized game form, 320 metrizable space, 84 graph (of a correspondence, 111 Michael's Selection Theorem, 309 Hahn-Banach Theorem, 207 Minkowski Functional, 248 Halpern-Bergman Theorem, 303 Minkowski's Inequality, 186 Hausdorff space, 79 natural subnets, 47 HOlder's Inequality, 183, 186 neighborhood, 5 homeomorphism, 21 of a set, 118 image of U under cp, 112 base, 238 input requirement net, 44 correspondence, 175 nested sequence (of sets), 35 interior point, 5 Nested Sphere Theorem, 35 interior (of a set), 5 norm of a linear function, 197 intersection correspondence, 157 normal space, 79 intersects ('U intersects X'), 8 nowhere dense, 28 inward-pointing, 302 open correspondence, 173 isometric, 221 open function, 21 isometry, 221 Open Mapping Theorem, 212 Jensen's Inequality, 188 open sections, 175 open sets, 2 Kakutani-Fan-Glicksberg Fixed- open sphere, 26 Point Theorem, 304 kernel (of a linear function), 245 paracompact space, 104 K-K-M correspondence, 299 Pareto efficient allocation, 256 K-K-M Theorem, 298 partition of unity, 107 subordinate to a family, 107 £.-(£1, £2), 198 existence of a, 108, 109 Lebesgue's Theorem, 91 point of accumulation, 10 Leontief model, 40 point of closure, 8 limit point, 10 pointwise bounded, 203 line segments, 98 product of correspondences, 162 linear isometry, 222 product space, 55,57 linear topological space, 99, 227 production correspondence, 175 linear topology, 99, 227 projection function, 56 fp-space, 185, 190 locally convex Hausdorff sp., 268 quasi-competitive equilibrium, 258 locally convex space, 263 refinement, 104 locally finite, 102 reflexive space, 225 lower hemi-continuous (lhc), 119 regular space, 79 INDEX 339

Riesz' Lemma, 217 triangulation, 288 selection from