Standard topology and discrete topology

[ideas]

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?

 

[Mathnerdmo]

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)

 

[Nessy]

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.