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  3. consider the following collections of subsets of r

Consider the following collections of subsets of R: B1 ={(a,?):a?R}, B2 ={(??,a):a?R}, B3 ={[a,?):a?R}, B4 = {[a, b] :...

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Advance Math

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 topologiesT1,...,T6, eg, are any of the topologies the same, are anycomparable, are any equal to familiar topologies on R, etc?

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