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- Show Corollary 1.24. Namely, show that a measure space (X, M , μ) is complete if (X, M , μ) is constructed via Carath ́eodory’s theo- rem (Theorem 1.20). Corollary 1.24. Let (X, M , μ) be a measure space obtained via Theo- rem 1.20. Then (X, M , μ) is complete. Theorem 1.20 (Carath ́eodory’s theorem). Let M be as above. We have (1) M is a σ-algebra.(2) ForE∈M,defineμ(E):=ν(E). ThenμisameasureonM.a. Is there any relation between reflexive normed space and a Banach Space? (If yes then prove) b. Give two examples of normed spaces that are not reflexive (with brief reasoning).This exercise demonstrates the concepts of boundary point, open and closed sets, etc., highly dependent on X's mother space. Give a reason for its correctness.Suppose Y=[ 0 ,2 ) . In this case A=[1,2 ) is not closed in X; while closed in Y. In addition, G=[0 ,1 )is not open in X while it is open in Y.Suppose X=R^2 and Y=R . In this case A=(0 ,1) is not open in X while it is open in Y. In fact, inside A in X is empty!
- Topology:Q13 or 14 For each of the following, if the statement about a topological space is always true, prove it; otherwise, give a counterexampleTrue or False: a)Every subset of a topological space is either open or closed.b)If X is a topological space with the discrete topology and if Xhas least two elements, then X is not connected.c) True or False: If X is a topological space, then there always is a metric on Xwhich gives rise to its topology.d) True or False: If X and Y are topological spaces and if f : X → Y is a constantmap (which means that there is a point y ∈ Y such that f(x) = y for all x ∈ X),then f is continuous.e) True or False: If X is a topological space, then X is both open and closedProve Corollary 5.5.10 - Let f be a continuous real-valued function defined on a metric space (X, d), and let D be a compact subset of X. Then f assumes maximum and minimum values on D.
- Topology For each of the following, if the statement about a topological space is always true, prove it; otherwise, give a counterexampleThe topic is boundedness and fields in calculus.1. Suppose that A is not an empty set, and is a set of real numbers bounded below. Let -A be the set of all numbers -x with x ∈ A. Prove that infA = -sup(-A).PS: inf means least upper bound and sup means the greatest lower bound.2. Suppose that 0 < s ∈ R and fix s. Put a,b,c,d ∈ Q where b,d > 0 and f = a/b = c/d such that sr = (sa )(1/b). Now if x ∈ R, define S(x) = {St : t ∈ Q and t less than or equal x} Prove that St = supS(f)PS: sup is the same as the greatest lower bound3. Prove that if the multiplicative axiom of a field implies that if x≠0 and xy = x, then y =1.Proof that R3 = W1 ⊕ W2, where W1 = {(x1, x2, x3) : x1 + x2 + x3 = 0} and W2 = Lin(1, 1, 1). Lin - space