Exploring Quotient Spaces: Visualizing Equivalence Classes

[shankarvn]

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?

 

[matt grime]

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.

 

[shankarvn]

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

 

[shankarvn]

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

 

[matt grime]

Forget the group stuff. Equivalence classes and relations come up in lots of mathematics, and that was another *example* of one.