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Hi; I'm trying desperately to understand many of these advanced principals of mathematics, such as discrete groups, but no matter how many times I review the material, it doesn't sink in. Could someone please provide examples, problems to solve (with their solutions) and/or ways to visualize this?
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I think you have already answered this, but just in case: "Discrete" means every set is open and every set is closed. Since arbitrary unions of open sets are open, to be discrete means that singletons are open. Since topological groups are "the same everywhere", it is enough to require that the
265:. Many people require all topological groups to be Hausdorff, so you are very right to say every singleton is closed. A discrete group is special: every singleton is both open and closed.
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Is the isomorphism to the category of discrete groups a functor and is it adjoint to the inverse isomorphism? Woud the inverse isomorphism be a forgetful functor?
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Messy introduction, the rest is a rather unbalanced list. The article should have a clear goal, or be split/merged with others (such as
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On the other hand, "Every singleton is closed" is a more natural property enjoyed by many more topological spaces, for instance every
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Would it be reasonable to say that a discrete group is one with a cardinality not greater than aleph-null?
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