22M:132: Topology Exam 1Sept 30, 2008[10] 1.) Definition: The point x ∈ X is a limit point of A if[12] 2.) Suppose R = the set of real numbers has the standard topology. Let Q = the set ofrational numbers. Calculate the following in R:Qo= Q = Q0=3.) The following two statements are false. Show that the statements are false by providing counter-examples. You do not need to explain your counter-examples.[9] 3a.) If xn∈ A, then there exists a unique point x ∈ A such that xn→ x.[9] 3b.) If f : X → Y is continuous, then f(A) ⊂ f (A).1[60] Prove 2 of the following 3. Clearly indicate your choices.1. The product of two Hausdorff spaces is Hausdorff.2. Let Y b e a subspace of X. If A ⊂ Y , then ClY(A) = A ∩ Y .3. Show that D((x1, y1), (x2, y2)) = max{|x1− y1|,|x2−y2|2} is a metric on R2where (xi, yi) ∈ R2.Show that this metric generates the standard topology on
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