Exploring Quotient Spaces: Visualizing Equivalence Classes

In summary: You can read more about it if you like, but it's not crucial to understanding the concept of a quotient space.
  • #1
shankarvn
13
0
Hi

I just wanted to know what a qoutient space is . Is there a physical picture to it? How can one imagine what an equivalence class,equivalence relation is?
 
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  • #2
The quotient space is the set of equivlance classes. How one pictures it, if one should even bother doing so depends on the context.

Given where you've posted this, I guess you mean things like:

Consider RxR with the relation (x,y)~(u,v) iff x-u and y-v are integers.

With experience, you instantly notice that is the torus.

How? Imagine the plane. We identify firstly all the x coordinates with the same non-integer component, and as we go from 0 to 1 we 'wrap' round again to 0, so that's like rolling the plane up into a big cylinder. similiary in the y direction we wrap the cylinder into itself.

Obviously for more complicated examples we can't even picture the initial space, never mind using that to construct the quotient space in our heads.


An equivalence class is the set of all points that are equivalent under an equivalence relation. again, experience is the best thing here.

An equivalence relation is the same thing as a partition of a set.

What are the equivalence classes of some group G when the relation is x~y if there is a z such that zx=yz?

Can you show that's an equivalence class?

It'd help to know what level of material you're used to.
 
  • #3
Hi

I am doing a course in differential geometry. I have an engg back ground. The concept of quotient spaces comes every now and then in our class. Your explanation does give a picture. I will read more and get back to you. Thanks
Shankar
 
  • #4
Hi
I did not follow whatever you said about groups. I have no clue as to what a group is though I have encountered that also before. I have not seen the group interpretation of equiv classes. I have seen equiv classes being defined in order to define a quotient space. Could you tell me what is the right order to take math courses. I do not or probably I cannot take too much of pure math. I need to apply this stuff in engg. Thanks
 
  • #5
Forget the group stuff. Equivalence classes and relations come up in lots of mathematics, and that was another *example* of one.
 

1. What is a quotient space?

A quotient space is a mathematical concept that refers to a space that is formed by partitioning a larger space into smaller subsets, known as equivalence classes. The quotient space consists of these equivalence classes, and each class represents a unique element of the original space.

2. How are equivalence classes visualized?

Equivalence classes are often visualized using a partitioning diagram, where each subset is represented by a distinct color or shape. Another way to visualize equivalence classes is by using a tree diagram, where each branch represents a different equivalence class.

3. What is the significance of exploring quotient spaces?

Exploring quotient spaces allows us to better understand the structure and relationships within a given space. It can also help us identify patterns and symmetries, and make connections between seemingly unrelated elements.

4. Can quotient spaces be applied to real-world problems?

Yes, quotient spaces can be applied to a variety of real-world problems, such as data analysis, image processing, and graph theory. They can also be used in fields like physics, biology, and social sciences to model complex systems and phenomena.

5. Are there any limitations to exploring quotient spaces?

One limitation of exploring quotient spaces is that it can be a complex and abstract concept, making it difficult to apply to certain problems without a strong mathematical foundation. Additionally, in some cases, the partitioning of a space into equivalence classes may not be unique, leading to different interpretations and results.

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