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
Y and g: X
Z are continuous. Prove the following function
Y X Z
when the codomain carries the product topology.
Exercise 0.18. Let (X,dx) and (Y,dy) be metric spaces. We can define a
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
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