However, extending this definition to correspondences is problematic, because the mean- ing of "f-(V) "is ambiguous. The following definition presents two alternatives Definition 3 Let(X, Ix) and(Y, Ty)be topological spaces, and consider a correspondence f: X=Y. For any V C Y, the upper inverse of f at V is f(v)=far: f(a)CV; the lower inverse of f at V is f(V)={x:f(x)∩V≠0} Clearly, fu(v)cf(v), and the two definitions coincide for singleton-valued dences. i.e. for functions Each notion gives rise to a corresponding definition of continuity Definition 4 Let(X, Tx) and(Y, Iy) be topological spaces, and consider a correspondence f: X>Y. Then f is upper hemicontinuous(uhc)iff, for every V E Ty, fu(V)EIx; f is lower hemicontinuous(lhc)iff, for every V E Ty, f(V)ETx; f is continuous iff it is both uhc and lhc Definition 4 highlights the connection with the notion of continuity for functions, but is somewhat hard to apply. However, the following characterization is useful Theorem 0.1 Let(X, Ix) be metrizable, and(Y, Ty) be compact and metrizable. Then a correspondence f: X=Yis (1)upper hemicontinuous iff, for every pair of convergent sequences aln>0 x in X and{y"}n≥0→ y in Y such that y∈f(x),y∈f(x); (2) lower hemicontinuous iff, for any sequence aIn>0-a in X, and for every E f(a there exists a subsequence ank k>o in X and a sequence y"k ]k>o in Y such that ynk E f(a"k for all k≥0, and ynk→y For infinite games, under our assumptions, the Nash best-reply correspondence is upper hemicontinuous as a consequence of the Maximum Theorem. However, in order to define Nash equilibrium for finite games(with the"trick"I mentioned above), a direct proof is easy to provide Existence of Nash Equilibrium To establish the desired result we need two ingredients: a"big mathematical hammer the Kakutani-Fan-Glicksberg Fixed-Point Theorem, and a trick. Let us start with the former. 2 B]For the more mathematically inclined, the domain of the correspondence in the next theorem need only first-countable 2For the more mathematically inclined, the theorem actually applies to locally convex hausdorff topo- logical vector spaces, such as e.g. the set of real-valued functions on a nonempty set X(endowed with the 3However, extending this definition to correspondences is problematic, because the mean￾ing of “f −1 (V )” is ambiguous. The following definition presents two alternatives. Definition 3 Let (X, TX) and (Y, TY ) be topological spaces, and consider a correspondence f : X ⇒ Y . For any V ⊂ Y , the upper inverse of f at V is f u (V ) = {x : f(x) ⊂ V }; the lower inverse of f at V is f ` (V ) = {x : f(x) ∩ V 6= ∅}. Clearly, f u (V ) ⊂ f ` (V ), and the two definitions coincide for singleton-valued correspon￾dences, i.e. for functions. Each notion gives rise to a corresponding definition of continuity: Definition 4 Let (X, TX) and (Y, TY ) be topological spaces, and consider a correspondence f : X ⇒ Y . Then f is upper hemicontinuous (uhc) iff, for every V ∈ TY , f u (V ) ∈ TX; f is lower hemicontinuous (lhc) iff, for every V ∈ TY , f ` (V ) ∈ TX; f is continuous iff it is both uhc and lhc. Definition 4 highlights the connection with the notion of continuity for functions, but is somewhat hard to apply. However, the following characterization is useful:1 Theorem 0.1 Let (X, TX) be metrizable, and (Y, TY ) be compact and metrizable. Then a correspondence f : X ⇒ Y is: (1) upper hemicontinuous iff, for every pair of convergent sequences {x n}n≥0 → x in X and {y n}n≥0 → y in Y such that y n ∈ f(x n ), y ∈ f(x); (2) lower hemicontinuous iff, for any sequence {x n}n≥0 → x in X, and for every y ∈ f(x), there exists a subsequence {x nk }k≥0 in X and a sequence {y nk }k≥0 in Y such that y nk ∈ f(x nk ) for all k ≥ 0, and y nk → y. For infinite games, under our assumptions, the Nash best-reply correspondence is upper hemicontinuous as a consequence of the Maximum Theorem. However, in order to define Nash equilibrium for finite games (with the “trick” I mentioned above), a direct proof is easy to provide. Existence of Nash Equilibrium To establish the desired result, we need two ingredients: a “Big Mathematical Hammer,” the Kakutani-Fan-Glicksberg Fixed-Point Theorem, and a trick. Let us start with the former.2 1For the more mathematically inclined, the domain of the correspondence in the next theorem need only be first-countable. 2For the more mathematically inclined, the theorem actually applies to locally convex Hausdorff topo￾logical vector spaces, such as e.g. the set of real-valued functions on a nonempty set X (endowed with the product topology). 3