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Let X = NN endowed with the product topology. For x ? X denote x by...

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Advance Math

Let X = NN endowed with the product topology. For x ?X denote x by (x1, x2, x3, . ..).

(a) Decide if the function given by d : X × X ? R is a metric onX where, d(x, x) = 0 and if x is not equal to y then d(x, y) = 1/nwhere n is the least value for which xn is not equal toyn. Prove your answer.

(b) Show that no compact set in X can contain a non-empty openset.

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