Elementary Concepts In Topology by Alexandroff P.

By Alexandroff P.

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I [a, b] is of interest, as the reader who has studied the mean value theorem in differential calculus knows. 10 Product spaces. 7. Given two metric spaces (X, d x) and (Y, d y) we can define several metrics on X x Y. For points (x 1 , yl) and y = (x2, Y2) in X x Y let d1 ((x1, yl), (x2, Y2)) = dx(xl, x2) + dy(y1, Y2), d2((x1, yl), (x2, Y2)) = [dx(xJ, x2) 2 +dy(Yl, Y2) 2 p, 1 doc((xl, YJ), (x2, Y2)) = max{dx(xl, x2), dy(y1, Y2)}. 16). 7 any one of these deserves to be called a product metric. We shall see in the next chapter that they are all equivalent in a certain sense.

10 Product spaces. 7. Given two metric spaces (X, d x) and (Y, d y) we can define several metrics on X x Y. For points (x 1 , yl) and y = (x2, Y2) in X x Y let d1 ((x1, yl), (x2, Y2)) = dx(xl, x2) + dy(y1, Y2), d2((x1, yl), (x2, Y2)) = [dx(xJ, x2) 2 +dy(Yl, Y2) 2 p, 1 doc((xl, YJ), (x2, Y2)) = max{dx(xl, x2), dy(y1, Y2)}. 16). 7 any one of these deserves to be called a product metric. We shall see in the next chapter that they are all equivalent in a certain sense. The definition may be extended to the product of any finite number of metric spaces.

X ~ ---t : X 0 X x X of any set X is the ---t X x X of any metric Proof As before we use the metric d 1 on X x X defined by d1((x1, x2), (x~, x~)) = dx(x,, x;) + dx (x2, Let c > 0. Put 8 = c/2. Then whenever dx(x, x') < 8 we have dt(~(x), ~(x')) = d1((x, x), (x', x')) = dx(x, x') This establishes continuity of ~. x~). + dx(x, x')

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