Exercise 0.13. Let X be a metric space and (xn) C X a sequence. Sho...

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Exercise 0.13. Let X be a metric space and (xn) C X a sequence. Show that (xn) converges to the point x € X if and only if the sequence of real numbers d(xn,x) converges to zero.Exercise 0.15. Given topological spaces X and Y, show that the (first) pro- jection map TT : X x Y X defined by (x,y) I is continuous Exercise 0.17. Given topological spaces X, Y and Z, suppose the functions f: X Y and g: X Z are continuous. Prove the following function is continuous: h: X Y X Z I (f(x),g(x)) when the codomain carries the product topology. Exercise 0.18. Let (X,dx) and (Y,dy) be metric spaces. We can define a function (xxY)² R as follows: = This is called the product metric. Prove it is indeed a metric, and that it generates the product topology. Thus show every product of metric spaces is a metric space.

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