Reference no: EM13996704

REAL ANALYSIS

Assignment:

1. If A and B are subsets of a topological space X, then A^- U B^- = (A U B)^- and A^° ∩ B^o = (A ∩ B)°, but the equalities A^- ∩ B^- = (A ∩ B)^- and A° U B° = (A U B)^o 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 {p(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) = p(a, b).

(b) There exist disjoint nonvoid closed subsets A and B of it 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 {p(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) = p(x, a). Is a unique?

(c) If X = R" 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|, R_eΦ and ImΦ are all continuous at p. If ψ(x) ≠ 0 ∀ x ∈ X, then 0/(1) 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 he two metric spaces where Y is complete, let D  ⊂ X be dense in X, and let f : D → Y be uniformly continuous on R. Then there exists g: X → Y that is uniformly continuous on X such that g(d) = f(d) for all d ∈ D. Pint: f maps Cauchy sequences to Cauchy sequences. If d_n → x ∈ X,  let g(x) = limn→∞f(d_n).]

8. Let f_n,(x) = x+ 1/n, f (x) = x for n ∈ N. Show that f_n → f uniformly on R, but it is false that f_n² → f² uniformly on R. of course f_n² → f² pointwise on R.