Group of translations on real line with discrete topology

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Main Question or Discussion Point

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?
 

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  • #2
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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?
 
  • #3
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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.
 
  • #4
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I don't understand your question,
OK, I didn't phrase that right. What I meant was that the real line has a discrete topology.
Yes- everything has a discrete topology.
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.
What are you defining as a "translation"?
 

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