HomeWork 1 DUE DATE : SEPT 28, 2010MATH 660 Instructor: Dr. RON SAHOODirection: This home work worths 75 points. Do all the problems. A small setof selected problems will be graded. In order to receive full credit, answer eachproblem completely and must show all work. Please write legibly.1. Prove the distributive law AS\γ∈ΓBγ=\γ∈ΓA[Bγfor union and intersections.2. Prove that A ⊆ B if and only if B0⊆ A0.3. Find the setlimn→∞inf An0and then justify your answer with a proof.4. Suppose that A1= A3= A5= · · · = A and A2= A4= A6= · · · = B. Then findlimn→∞sup Anand limn→∞inf Anand then justify your answer with a proof.5. Let A1, A2, A3, ... be a sequence of non-empty sets such that An⊆ An+1for each n ∈ N.Find limn→∞sup Anand limn→∞inf An. Give an example of such a sequence.6. Let A1, A2, A3, ... be a sequence of non-empty sets such that An+1⊆ Anfor each n ∈ N.Find limn→∞sup Anand limn→∞inf An. Give an example of such a sequence.7. Let A1, A2, A3, ... and B1, B2, B3, ... be any two sequences of non-empty subsets of agiven set X. Prove that limn→∞sup(An∪ Bn) =limn→∞sup An[limn→∞sup Bn.8. Let A1, A2, A3, ... and B1, B2, B3, ... be any two sequences of non-empty subsets of agiven set X. Prove that limn→∞inf(An∩ Bn) =limn→∞inf An\limn→∞inf Bn.9. Give an example of an ordered pair (Xτ) which is a topological space but for which τ isnot a σ-algebra. Justify your answer.10. Give an example of a non-empty set X and a collection C of subsets of X such that Cis a σ-algebra but (X, C) is not a topological space. Justify your answer.11. Give an example of a non-empty set X and a collection τ of subsets of X such that τis a a topological space and a σ-algebra. Justify your answer.12. Let f : X → Y . Let A ⊆ X and B ⊆ Y . Further, define f (A) to be the set{f(x) ∈ Y | x ∈ A}.(a) Prove that A ⊂ f−1( f(A) )(b) Show that in general A 6= f−1( f(A) )(c) Show that ff−1(B)⊆ B.13. Let X = R. Let H be the collection of all non-empty subsets U of R such that foreach point p ∈ U there exists an interval of the form [a, b), a < b such that p ∈ [a, b) and[a, b) ⊆ U. Let τ1= {∅} ∪ H. Prove that (X, τ1) is a topological space.14. Let R be the set of reals and let τ1= P(R), the collection of all subsets of R. Further,let τ2= { ∅, R }. It is easy to prove that (X, τ1) and (X, τ2) two topological spaces. Let fbe a mapping on R defined asf(x) =(1 if x ≥ 0−1 if x < 0.(a) Is f : (R, τ1) → (R, τ2) continuous? (b) Is f : (R, τ2) → (R, τ1) continuous? Justify youranswer.15. Let R be the set of reals and let τ1= P(R), the collection of all subsets of R. Further,let τ2= { ∅, R }. It is easy to prove that (X, τ1) and (X, τ2) two topological spaces. Let fbe a mapping on R defined asf(x) =(0 if x ∈ Q1 if x ∈ Q0.(a) Is f : (R, τ1) → (R, τ2) continuous? (b) Is f : (R, τ2) → (R, τ1) continuous? Justify youranswer.September 2,