Standard topology and discrete topology

In summary, the conversation revolves around comparing 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. The speaker asks about the basic open sets in the discrete topology and which of these are open in the topology generated by S. They also mention that the topology generated by S is different from the standard topology. Another person asks about the relationship between topologies on a set and its power set.
  • #1
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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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  • #2
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)
 
  • #3
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.
 

1. What is the difference between standard topology and discrete topology?

The standard topology is the most commonly used topology in mathematics and is defined by open sets, while the discrete topology is the most basic topology and is defined by every subset being an open set.

2. How do you determine the open sets in a standard topology?

In a standard topology, the open sets are typically determined by a basis, which is a collection of sets that satisfies certain properties. These sets can then be used to generate all other open sets in the topology.

3. What are the advantages of using a discrete topology?

The discrete topology has the advantage of being easy to understand and work with, making it a good starting point for understanding more complex topologies. It is also useful in certain applications, such as computer science and digital signal processing.

4. How are continuous functions defined in standard topology and discrete topology?

In a standard topology, a function is continuous if the inverse image of an open set is also an open set. In a discrete topology, all functions are continuous since every subset is an open set.

5. Can a space have both a standard and a discrete topology?

Yes, a space can have multiple topologies defined on it. It is possible for a space to have both a standard topology and a discrete topology, as well as many other topologies. These topologies may have different properties and characteristics, allowing for different types of analysis and applications.

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