Group of translations on real line with discrete topology

In summary, the conversation discussed the differences between the group of translations on a real line with discrete topology and the group of translations on a real line with the usual topology. It was noted that both groups satisfy all group axioms and are topological groups since all self-bijections are continuous. The possibility of Td being a Lie group was also mentioned, with the clarification that it would be a 0-dimensional Lie group. The conversation also touched on the concept of a "generator" for the real line, as well as the question of whether the two groups of translations mentioned are actually the same.
  • #1
xboy
134
0
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
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
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
I don't understand your question,
xboy said:
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.
xboy said:
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"?
 
  • #5


The group of translations on a real line with discrete topology, Td, is different from the group of translations on a real line with the usual topology, Tu, in several ways. Firstly, the discrete topology on the real line means that every subset is open, so the group Td will have a different topology than Tu, which has a continuous topology. This means that the group operations, such as multiplication and inversion, will also be different in Td compared to Tu.

Furthermore, Td will not be a Lie group, as a Lie group requires a smooth manifold structure, which is not present in Td due to the discrete topology. In contrast, Tu does have a smooth manifold structure and is therefore a Lie group.

In terms of generators, Td and Tu may have different sets of generators. The generators of Td will depend on the specific translations chosen, while the generators of Tu will be related to the continuous translations along the real line.

Overall, Td and Tu are distinct groups with different properties and structures due to their different topologies. It is important to note that even though they may have some similarities, they are fundamentally different groups.
 

1. What is a group of translations on the real line with discrete topology?

A group of translations on the real line with discrete topology is a mathematical structure that consists of a set of real numbers and a binary operation of addition. This group represents the set of all possible translations or shifts on the real line, where each element of the group represents a specific translation by a fixed amount.

2. What is the discrete topology on the real line?

The discrete topology on the real line is a way of defining the open sets on the real line. In this topology, every point on the real line is an open set, and any union of these points is also an open set. This topology is different from the usual topology on the real line, which includes open intervals and open rays.

3. How is the group of translations on the real line with discrete topology different from other groups?

The group of translations on the real line with discrete topology is different from other groups because it has a specific structure and properties that are unique to this particular set of translations. In this group, the operation of addition is commutative, and every element has an inverse, making it an abelian group. It also has a finite number of elements, unlike other infinite groups.

4. What are the applications of this group in science?

This group has various applications in science, particularly in the fields of physics and mathematics. In physics, this group is used to study the symmetries of physical systems, such as the translation invariance of a particle in space. In mathematics, it is used to study abstract algebra and group theory, which has applications in cryptography, coding theory, and other areas.

5. Are there any real-world examples that can be modeled using this group?

Yes, there are real-world examples that can be modeled using this group. For instance, if we consider a system of particles moving on a line, the set of all possible translations or shifts of these particles can be represented by this group. Similarly, the movements of objects in a discrete grid or lattice can also be modeled using this group of translations on the real line with discrete topology.

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