2. Suppose (X, dx), (Y, dy) are metric spaces. The product metric of X X Y...

 

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2. Suppose (X, dx), (Y, dy) are metric spaces. The product metric of X X Y is the function d((x1, y), (2, y2)) := dx(x, x) + dy (y, y2). (a) Show that (X Y, d) is indeed a metric space. Hint: For the triangle inequality, argue as follows. If x1, x2, x3 X and Y1, Y2, Y3 Y, let uo (dx(x1, x3), dy (y, y3)), (dx(x1, x), dy (y, y2)), = (dx(x2, 23), dy (2, 3)). = Then ou + 2| |1| + ||. (b) Show that every open set of X x Y is a union of sets of the form U V where U is open in X and V is open in Y. (c) Let E C X and F C Y be compact. Using the open cover definition of compactness, show that Ex F is a compact subset of X x Y. Hint: Suppose Ex C Ux,y (Ux,y Vx,y) with Ux,y C X and Vx,y C Y open. For each fixed x E, {Vx,y}yeF is an open cover of F, so it has a finite subcover, say {Vx,y,j}j. Now Ux := NjUx,y,j is open and {U}xE is an open cover of E. (d) Suppose E,..., Ek C R are compact. Using (c) and induction, show that E x ECR is compact. Remark: There are in fact many product metrics, though this one is especially intuitive. Part (b) shows that the product topology of X and Y agrees with the topology of the product metric. 2. Suppose (X, dx), (Y, dy) are metric spaces. The product metric of X X Y is the function d((x1, y), (2, y2)) := dx(x, x) + dy (y, y2). (a) Show that (X Y, d) is indeed a metric space. Hint: For the triangle inequality, argue as follows. If x1, x2, x3 X and Y1, Y2, Y3 Y, let uo (dx(x1, x3), dy (y, y3)), (dx(x1, x), dy (y, y2)), = (dx(x2, 23), dy (2, 3)). = Then ou + 2| |1| + ||. (b) Show that every open set of X x Y is a union of sets of the form U V where U is open in X and V is open in Y. (c) Let E C X and F C Y be compact. Using the open cover definition of compactness, show that Ex F is a compact subset of X x Y. Hint: Suppose Ex C Ux,y (Ux,y Vx,y) with Ux,y C X and Vx,y C Y open. For each fixed x E, {Vx,y}yeF is an open cover of F, so it has a finite subcover, say {Vx,y,j}j. Now Ux := NjUx,y,j is open and {U}xE is an open cover of E. (d) Suppose E,..., Ek C R are compact. Using (c) and induction, show that E x ECR is compact. Remark: There are in fact many product metrics, though this one is especially intuitive. Part (b) shows that the product topology of X and Y agrees with the topology of the product metric.