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Standard topology and discrete topology

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How to compare the topology on R generated by the subbasis S={[x,y)|x,y are rational}U{(x,y]|x,y rational} to the discrete topology on R?
 

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Well, what are the basic open sets in the discrete topology?

Which of these are open in the topology generated by S and which aren't? (by the way, the topology generated by S is far from the standard topology)
 
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How to compare the topology on R generated by the subbasis S={[x,y)|x,y are rational}U{(x,y]|x,y rational} to the discrete topology on R?
On any given set X the discrete topology is the whole power set of X. How are topologies on X related to the power set of X? I hope you get it.
 

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