Consider the following collections of subsets of R: B1 ={(a,∞):a∈R}, B2 ={(−∞,a):a∈R}, B3 ={[a,∞):a∈R}, B4 =...

Consider the following collections of subsets of R:

B1 ={(a,∞):a∈R},

B2 ={(−∞,a):a∈R},

B3 ={[a,∞):a∈R},
B4 = {[a, b] : a, b ∈ R},
B5 = {[a, b] : a, b ∈ Q},
B6 ={[a,b]:a∈R,b∈Q}.

(i) Show that each of these is a basis for a topology on R.

(ii) What can you say about the corresponding topologies T1,...,T6, eg, are any of the topologies the same, are any comparable, are any equal to familiar topologies on R, etc?

Expert Solution

Colby Messinger answered 2 years ago

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