Let ℊ = {N(p; r): p, r ∈ ℚ and r > 0}.
(a) Prove that ℊ is countable.
(b) Let A be a nonempty open set and let ℊA = {N ∈ ℊ: N ⊆ A}. Prove that ∪{N: N ∈ ℊA} = A. What is the cardinality of ℊA?
(c) Let ℱ be any nonempty collection of nonempty open sets. Prove that there is a. family ℊℱ ⊆ ℊ such that ∪ {N: N ∈ ℊℱ) = ∪ {F:F ∈ ℱ}. Then use ℊℱ to show that there is a countable subfamily ℋ ⊆ ℱ such that ∪{H ∈ ℋ } = ∪ {F ∈ ℱ }.
(d) Prove the Lindelof covering theorem: Let S be a subset of ℝ and let ℱ be an open covering of S. Then there is a countable subfamily of ℱ that also covers S.
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