Isomorphic between plane and line

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In summary, the conversation discusses the possibility of an isomorphism between the open interval (0,1) and the real line \mathbb{R}^1, as well as between \mathbb{R}^2 and \mathbb{R}^1. The experts clarify that there is no isomorphism between \mathbb{R}^2 and \mathbb{R}^1 as a topological space, but there are Borel and group isomorphisms between the two. However, there is no map that is both a Borel and group isomorphism between \mathbb{R}^2 and \mathbb{R}^1.
  • #1
hermitian
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Hi,

I understand that the open interval (0,1) is isomorphic to the real line [tex]\mathbb{R}^1[/tex]

May i know whether there is also isomorphism from [tex]\mathbb{R}^2[/tex] to [tex]\mathbb{R}^1[/tex]

thanks a lot!
 
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  • #2
What do you mean by an isomorphism in this case?
 
  • #3
hmm... the usual sense of isomorphism... where there is a structure preserving bijective map from R^2 to R^1. sorry is my question still vague?
 
  • #4
The question is which structure are you interested in! It's structure as a set? As an additive group? As a topological space?

I imagine it's the topology you care about. In that case, the answer is "no". The common proof (AFAIK) is to consider the image of a subset of R2 in the shape of the letter Y -- if such an isomorphism existed, it must be injective on the Y. Can you derive a contradiction from that?
 
  • #5
There is a Borel isomorphism between [itex]\mathbb{R}^2[/itex] and [itex]\mathbb{R}[/itex].

There is a group somorphism between [itex]\mathbb{R}^2[/itex] and [itex]\mathbb{R}[/itex] (both with addition).

BUT: There is no map between [itex]\mathbb{R}^2[/itex] and [itex]\mathbb{R}[/itex] that is both a Borel isomorphism and a group isomorphism.
 

FAQ: Isomorphic between plane and line

What does it mean for a plane and a line to be isomorphic?

Isomorphism refers to a specific type of mapping between two mathematical structures that preserves their structure and relationships. In the case of a plane and a line, an isomorphic relationship means that there is a one-to-one correspondence between the points of the plane and the points of the line, such that the geometric properties of both structures are preserved.

How can you show that a plane and a line are isomorphic?

To prove that a plane and a line are isomorphic, you need to demonstrate that there exists a one-to-one mapping between the points of the plane and the points of the line that preserves distances, angles, and other geometric properties. This can be done through various methods, such as using coordinate systems or transformations like rotations and translations.

What is the significance of an isomorphic relationship between a plane and a line?

An isomorphic relationship between a plane and a line is significant because it allows us to study and understand the properties of one structure by looking at the properties of the other. This can be especially useful in solving geometric problems or in the field of linear algebra, where isomorphic structures can be used to simplify calculations.

Are there any real-life applications of isomorphism between a plane and a line?

Yes, there are many real-life applications of isomorphism between a plane and a line. For example, in computer graphics, the concept of isomorphism is used to create 3D images on a 2D screen. In physics, isomorphic relationships between different mathematical structures are used to model physical phenomena and make predictions.

Can a plane and a line be isomorphic in more than one way?

Yes, it is possible for a plane and a line to be isomorphic in more than one way. This means that there can be multiple one-to-one mappings between the points of the plane and the points of the line that preserve their geometric properties. However, it is important to note that all isomorphic relationships between a plane and a line must preserve the basic structure and properties of both structures.

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