Real analysis hw

HW 8: MA 34100. DUE 3/29.

1. Suppose X,Y,Z are metric spaces, and Y is compact. Let f be a function mapping X into Y , g a continuous one-to-one mapping of Y into Z, and set h(x) = g(f(x)), for x ∈ X. Show that: (i) if h is uniformly continuous, then f is uniformly contin- uous; (ii) if h is continuous, then f is continuous; (iii) can the compactness of Y be omitted from the hypothesis to deduce (i) and (ii)?.

2. If E is a non-empty subset of a metric space X, define the distance from X to E via ρE(x) = infz∈E d(x,z). Show that: (i) ρE(x) = 0 if and only if x ∈ E; (ii) |ρE(x) −ρE(y)| ≤ d(x,y) for all x ∈ X and y ∈ Y ; (iii) ρE is uniformly continuous on X.

3.. Let X be a metric space. Suppose K ⊂ X is compact and F ⊂ X is closed and K ∩F = ∅. Show that: (i) there exists δ > 0 such that d(p,q) > δ if p ∈ K, q ∈ F; (ii) the conclusion in (i) may fail for two disjoint closed sets if neither is compact.

4. For A,B ⊂ Rn, if A is compact and B is closed, prove that A + B is closed (recall that A + B = {a + b : a ∈ A,b ∈ B}).

5. Let f be a continuous real-valued function on a metric space X. Let Z = {x ∈ X : f(x) = 0}. Prove that Z is closed.

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