1. Find an uncountable open cover f of R such that f has no finite subcover. Does f contain a countable subcover? 2. Let g = {N ( p; r) : p, r ∈  and r > 0}. (a) Prove that g is countable (b) Let A be...

1. Find an uncountable open cover f of R such that f has no finite subcover. Does f contain a countable subcover?

2. Let g = {N ( p; r) : p, r ∈
 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 ∈ g_A
} = A. What is the cardinality of g_A?

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

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