Reference no: EM132191441
REAL ANALYSIS
Assignment
1. If A and B are subsets of a topological space X, then A- ∪ B- = (A ∪ B)- and A? ∩ B? = (A ∩ B)?, but the equalities A- ∩ B- = (A ∩ B)- and A? ∪ B? = (A ∪ B)? may fail. Prove the first two equalities and give an example of two subsets of R for which both of latter two equalities fail.
2. Let X be a metric space and let A and B be nonvoid subsets of X. Define
dist(A, B) = inf{ρ(x, y) : x ∈ A, y ∈ B}.
Then
(a) If A and B are compact, then there exist a∈A and b∈B such that dist(A, B) = ρ(a, b).
(b) There exist disjoint nonvoid closed subsets A and B of R for which
dist(A, B) = 0.
3. Let X be a metric space and let A and B be nonvoid subsets of X. Define dist(x, A) = inf{ρ(x, a) : a ∈ A}.
(a) For x ∈ X; x ∈ A- if and only if dist(x, A) = 0.
(b) If A is compact and x ∈ X, then there exists an a A such that dist(x, A) = ρ(x, a). Is a unique?
(c) If X = Rn and A is closed, the the conclusion of (b) holds.
(d) It can happen that x ∈ X, A is closed, and dist(x, A) < ρ(x, a) for all a ∈ A.
4. In a Hausdorff Space, a sequence can converge to at most one point.
5. Let X be a topological space, let p ∈ X, and let φ and ψ be C-valued functions on X that are continuous at p. Then the functions φ + ψ, φψ, |φ|, Reφ and Imφ are all continuous at p. If ψ(x) ≠ 0 ∀ x ∈ X, then φ/ψ is also continuous at p.
6. Suppose that f and g are continuous functions from a topological sapce X into a Hausdorf space Y and that f (d) = g(d) for all d ∈ D, where D is a dense subset of X. Then f (x) = g(x) for all x ∈ X.
7. Let X and Y be two metric spaces where Y is complete, let D ⊂ X be dense in X, and let f : D → Y be uniformly continuous on D. Then there exists g : X → Y that is uniformly continuous on X such that g(d) = f (d) for all d ∈ D. [Hint: f maps Cauchy sequences to Cauchy sequences. If dn → x ∈ X, let g(x) = limn→∞ f (dn). ]
8. Let fn(x) = x + 1/n, f (x) = x for n ∈ N. Show that fn → f uniformly on R, but it is false that fn2 → f2 uniformly on R. of course fn2 → f2 pointwise on R.