Math 447: Real Variables Homework 7

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Homework 7

Math 447: Real Variables

Exercise 1 For each of the following subsets, calculate the interior, boundary and closure and determine whether the set is compact:

(a) Q ⊂ R.

(b) S2 (1) ∩ ()3 ⊂ R3, where S2 (1) and are as defined in Homework 6.

(c) The unit disc D3 ⊂ R3, where

D3 = { x ∈ R3 | d2(x, 0) ≤ 1 }.

(d) The plane { (x, y, z) | z = 0 } ⊂ R3

Exercise 2 Let X be a metric space. Recall that if A ⊂ B ⊂ X then A is dense in B if for all open subsets O ⊂ X if O ∩ B ≠ ∅ then O ∩ A ≠ ∅. Prove that A is dense in B if and only if B ⊂ A.

Exercise 3 Let X, Y be metric spaces and f : X → Y . Prove that the following four conditions are equivalent:

(a) f is continuous;

(b) f−1 (IntA) ⊂ Int(f −1 (A)) for all subsets A ⊂ Y ;

(c) f−1 (A) ⊃ f −1(A) for all subsets A ⊂ Y ;

(d) f(A) ⊂ f(A) for all subsets A ⊂ X;

Exercise 4

(a) A topological basis for a metric space is a collection B of open sets such that every open set is the union (possibly uncountable) of sets in B: (∀O open)(∃B' ⊂ B)

Find a basis B for R2 that has only countably many sets in it. Prove that every open subset of R 2 can be written as a union of sets in your basis.

Note: A topological space with a countable basis is called “second countable”.

(b) Let X be any metric space such that there is a countable basis B. Prove that X is separable.