Problem

Let ℊ = {N(p; r): p, r ∈ ℚ and r > 0}.(a) Prove that ℊ is countable.(b) Let A be a none...

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: NA}. Prove that ∪{N: N ∈ ℊA} = A. What is the cardinality of ℊA?


(c) Let be any nonempty collection of nonempty open sets. Prove that there is a. family ℊ  ⊆ ℊ such that ∪ {N: N ∈ ℊ) = ∪ {F:F}. Then use ℊ  to show that there is a countable subfamily ℋ ⊆ such that ∪{H ∈ ℋ } = ∪ {F }.


(d) Prove the Lindelof covering theorem: Let S be a subset of ℝ and let be an open covering of S. Then there is a countable subfamily of that also covers S.

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Solutions For Problems in Chapter 3.14S