Homeomorphism between a 1-dim vector space and R

In summary, the conversation discusses the concept of finding a homeomorphism between a 1-dimensional vector space and R, independent of the basis. There is discussion of defining the norm and inducing a topology on the vector space, as well as the difficulty in finding a basis-free homeomorphism. The conversation concludes with the suggestion of using a particular basis to define an isomorphism and the acknowledgement that this approach may still rely on the chosen basis.
  • #1
jem05
56
0
im trying to get a homeomorphism between a 1-dim vector space and R, but independent of the basis.
Any ideas?
 
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  • #2


Do you mean homomorphism? Homeomorphisms are maps between topological spaces.
 
  • #3


no i meant homeomorphism,
im trying to get a chart (manifolds- C^(/infinity) structure) from V to R
we can though define the norm right and induce a topology on our vector space,
but i still can't see the homeomorphism,
ofcourse the hard thing is that i want it to be independent of the basis, or else, it would've been trivial.
 
  • #4


I would be pretty surprised if you could find one that's basis free.

You can't even define the norm without a basis unless you have a basis free definition for a linear isomorphism between V and R.
 
  • #5


Can you give a precise statement of what you mean by "independent of basis"?
 
  • #6


yeah sure, i think they mean for whatever choice of bases, we can get the same result.
 
  • #7


Then pick a particular basis, and use that to define an isomorphism of vector spaces.

Once you have done that, you have defined a function. The value of the function depends only on the point chosen and nothing else.

(Of course, the obvious method for computing values of this function would probably make use of the basis you originally chose)
 

1. What is a homeomorphism?

A homeomorphism is a mathematical concept that describes a one-to-one and onto mapping between two topological spaces. It is a type of transformation that preserves the basic structure of the space, such as continuity and connectedness.

2. How does a 1-dim vector space relate to R (the set of real numbers)?

A 1-dim vector space is a mathematical structure that consists of all possible linear combinations of a single vector. R, on the other hand, is a set of all real numbers. A homeomorphism between these two structures means that there is a one-to-one correspondence between the elements of the vector space and the real numbers.

3. Can you give an example of a homeomorphism between a 1-dim vector space and R?

Yes, one example is the function f(x) = x. This function maps every element in the 1-dim vector space to its corresponding real number in R. It is a homeomorphism because it is continuous, one-to-one, and onto.

4. What are the implications of a homeomorphism between a 1-dim vector space and R?

The existence of a homeomorphism between these two structures means that they share similar topological properties. This allows us to use the concepts and techniques from one structure to understand and solve problems in the other. It also makes it easier to visualize and manipulate objects in the 1-dim vector space by using the familiar structure of R.

5. How is the concept of a homeomorphism used in real-world applications?

Homeomorphisms have various applications in fields such as computer science, physics, and engineering. In computer graphics, they are used to create smooth transformations and deformations of objects. In physics, they help describe the relationship between different physical systems. In engineering, they are used to model and analyze complex systems and networks.

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