Analysis

HW3: MA 34100. DUE 2/1.

1. For x, y ∈ R, define d1(x, y) = (x − y)2; d2(x, y) = √ |x−y|; d3(x, y) = |x2 − y2|;

d4(x, y) = |x − 2y|; d5(x, y) = |x−y|

1+|x−y|. For all i ∈ {1, . . . , 5}, determine whether di is a metric.

2. Let X be an infinite set. For p, q ∈ X, define d(p, q) = 1, if p 6= q, and d(p, q) = 0 otherwise. Which subsets of the corresponding metric space are open? Which are closed? Which are compact?

3. Prove that Rn is separable (a set is said to be separable if it contains a countable dense subset).

4. A collection {Vα} of open subsets of X is said to be a base for X if for every x ∈ X and every open set G ⊂ X such that x ∈ G, we have x ∈ Vα ⊂ G. Prove that every separable metric space has a countable base.

5. Let E = {1 + 1 n

: n ∈ N}∪{1}. Prove that E is compact directly from the definition.

6. Give an an example of an open cover for (−1, 1) ∪ (2, 3) with no finite subcover.

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