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Group of translations on real line with discrete topology

  1. Nov 25, 2011 #1
    Hi.

    I wanted to know in what way the group of translations on a real line with discrete topology (let's call it Td) will be different from the group of translations on a real line with the usual topology (lets call it Tu)? Is Td a Lie Group? Will it have the same generator as Tu?
     
  2. jcsd
  3. Nov 26, 2011 #2
    Note that every self-bijection will be continuous, which is rather boring.

    It will still satisfy all the group axioms- they don't depend on topology. It will be a topological group since everything will be continuous. It will be a Lie group, if you count 0-dimensional Lie groups, and it will have uncountably many disconnected components.

    All of the above can be said for any group, whether or not it already has a topology. Any group G can be considered a topological group simply by giving it the discrete topology (or a Lie group, albeit a 0-dimensional one).

    I don't know what you mean by "the generator of Tu". The real line doesn't have a generator, does it?
     
  4. Nov 26, 2011 #3
    OK, I didn't phrase that right. What I meant was that the real line has a discrete topology. Now I take the group of translations on it. My question was whether this group would be any different from the group of translations on a (real line with usual topology) and I think that they would be the same.
     
  5. Nov 27, 2011 #4
    I don't understand your question,
    Yes- everything has a discrete topology.
    What are you defining as a "translation"?
     
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