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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?
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?
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.
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.
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.
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.
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.