Thread: Quotient spaces and equivalence relations

Let I(x) denote the integer part of a real number x (ie the unique integer n such that ).

I can show that ~ iff is an equivalence relation.

Let ~ be the quotient map, let ~ be endowed with the quotient topology, and let U be an open set in ~. Prove that if is such that then .

Deduce that the open sets in ~are the empty set, ~ and the image sets where .

Consider the map by . Is the map I continuous (when Z is endowed with the induced topology)?

Originally Posted by Amanda1990

Let I(x) denote the integer part of a real number x (ie the unique integer n such that ).

I can show that ~ iff is an equivalence relation.

Let ~ be the quotient map, let ~ be endowed with the quotient topology, and let U be an open set in ~. Prove that if is such that then .

If then which, by the definition of the quotient topology, is an open set in R and therefore contains (for small enough). Hence .

Originally Posted by Amanda1990

Consider the map by . Is the map I continuous (when Z is endowed with the induced topology)?

Notice that as . But and p(1) = 1.

Thanks a lot - for the first part of the question I had the right idea but just couldn't write it properly.

For the last point you made, did you mean to use I instead of p? The map p just takes any real number to its equivalence class right? ie isn't ?

On a similar note, I agree that I is not continuous, but I does define a bijection, J say, from ~ (I think this i fairly clear). What is the topology on Z making J a homomorphism?

Originally Posted by Amanda1990

Thanks a lot - for the first part of the question I had the right idea but just couldn't write it properly.

For the last point you made, did you mean to use I instead of p? The map p just takes any real number to its equivalence class right? ie isn't ?

On a similar note, I agree that I is not continuous, but I does define a bijection, J say, from ~ (I think this i fairly clear). What is the topology on Z making J a homomorphism?

Yes, I should have said I rather than p.

The topology on Z making J a homeomorphism is given by the middle part of the question: The open sets are the intervals (–∞,n], together with the empty set and the whole space.

Well this is it - it looks as though the middle part of the question should give the answer to this but surely a topology on Z must be a family of subsets of Z (by definition). Then intervals (–∞,n] are not subsets of Z and so cannot form a topology (?)

Originally Posted by Amanda1990

Well this is it - it looks as though the middle part of the question should give the answer to this but surely a topology on Z must be a family of subsets of Z (by definition). Then intervals (–∞,n] are not subsets of Z and so cannot form a topology (?)

They certainly have to be subsets of Z, so you should interpret (–∞,n] as meaning . I suppose I should have written it as .