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

Let X = N^N 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 on X where, d(x, x) = 0 and if x is not equal to y then d(x, y) = 1/n where n is the least value for which x_n is not equal to y_n. Prove your answer.

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

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Expert Solution

Colby Messinger answered 10 months ago

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