Simple question relating diffeomorphisms and homeomorphisms.

In summary, the conversation discusses the relationship between diffeomorphisms and homeomorphisms in a Euclidean space or manifold. It is confirmed that every diffeomorphism is also a homeomorphism, but the reverse is not necessarily true. The need for a differentiable structure in order for a diffeomorphism to exist is also mentioned. The conversation ends with a humorous analogy about understanding the meaning of words in a question.
  • #1
Pinu7
275
5
Consider a Euclidean space or a manifold or whatever. Furthermore, consider two regions on this space. If one can construct a diffeomorphism between the points from one region to the other, does this imply that the two regions are homeomorphic?

My gut feeling is "yes," but I would like a confirmation with maybe an explanation.
 
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  • #2
Every diffeomorphism is in particular a homeomorphism, yes.
 
  • #3
though clearly not the converse. To have a diffeomorphism you need some sort of differentiable structure which an arbitrary topological space does not have.
 
  • #4
if i have ham and eggs, does that mean i have eggs? i.e. you could only ask this question if you do not know what the words in it mean.
 
  • #5
The explanation is the basic fact that a differentiable function is continuous.
 
  • #6
mathwonk said:
if i have ham and eggs, does that mean i have eggs? i.e. you could only ask this question if you do not know what the words in it mean.
Ha, nice answer :)
 

FAQ: Simple question relating diffeomorphisms and homeomorphisms.

What are diffeomorphisms and homeomorphisms?

Diffeomorphisms and homeomorphisms are both types of mathematical functions that preserve certain properties of a space. Diffeomorphisms are smooth and differentiable, while homeomorphisms are continuous. Both types of transformations preserve distances and angles between points in a space.

What is the difference between diffeomorphisms and homeomorphisms?

The main difference between diffeomorphisms and homeomorphisms lies in their level of smoothness and differentiability. Diffeomorphisms are smooth and differentiable, while homeomorphisms are only required to be continuous. This means that diffeomorphisms preserve more geometric properties of a space than homeomorphisms do.

How are diffeomorphisms and homeomorphisms used in mathematics?

Diffeomorphisms and homeomorphisms are used in mathematics to study and understand the properties of different spaces. They are particularly useful in topology and differential geometry, where they help to classify and compare different types of spaces.

What are some real-life applications of diffeomorphisms and homeomorphisms?

Diffeomorphisms and homeomorphisms have many real-life applications, particularly in the fields of physics, engineering, and computer graphics. For example, they are used in fluid dynamics to describe the motion of fluids, in robotics to study the movement of robots, and in computer graphics to generate realistic 3D images.

How do diffeomorphisms and homeomorphisms relate to each other?

Diffeomorphisms and homeomorphisms are both types of transformations that preserve certain properties of a space. Homeomorphisms are a broader class of transformations, as they only require continuity, while diffeomorphisms are a subset of homeomorphisms that also require differentiability. In other words, all diffeomorphisms are homeomorphisms, but not all homeomorphisms are diffeomorphisms.

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