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)?
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?
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 (?)