Standard topology and discrete topology

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SUMMARY

The discussion focuses on comparing the topology on R generated by the subbasis S = {[x,y) | x,y are rational} ∪ {(x,y] | x,y are rational} with the discrete topology on R. It establishes that the discrete topology consists of the entire power set of R, while the topology generated by S is significantly different from the standard topology. The participants analyze which open sets in the discrete topology are also open in the topology generated by S, highlighting the fundamental differences between these topological structures.

PREREQUISITES
  • Understanding of basic topology concepts, including open sets and subbases.
  • Familiarity with the discrete topology and its properties.
  • Knowledge of the standard topology on R.
  • Ability to work with rational numbers in the context of topology.
NEXT STEPS
  • Study the properties of the discrete topology in detail.
  • Explore the concept of subbases and their role in generating topologies.
  • Investigate the standard topology on R and its open sets.
  • Learn about the relationship between topologies and power sets in set theory.
USEFUL FOR

Mathematicians, students of topology, and anyone interested in understanding the differences between various topological structures on real numbers.

ideas
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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)
 
ideas said:
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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