Question 3 Recall that the metric space (M,d) is separated, in the sense that for all X,Y EM there are two open sets U a

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Question 3 Recall That The Metric Space M D Is Separated In The Sense That For All X Y Em There Are Two Open Sets U A 1 (46.79 KiB) Viewed 83 times

Question 3 Recall that the metric space (M,d) is separated, in the sense that for all X,Y EM there are two open sets U and V such that 2 EU, Y EV, and U NV = . Let 2 € M and B a nonempty subset of M such that x E B. (a) Can one find two open sets U and V such that I EU, B C V and U n V = 0? Explain. (b) Suppose now that a ¢ B an set 8 = d(x,B) := infyeB d(x, y). Show that U = B(x, ) and V = Uyeb B(y, ) are two open sets separating r and B. That is, r EU, BCV, and Un V = 0. ( 10, 25 Marks)