Recall that the integer part (or integral part) of a real number x is the unique integer n ∈ Z such that n ≤ x < n + 1. We denote it by I(x).

On R we deﬁne the relation xRy ⇔ I(x) = I(y).

(a) Let p : R → R/R be the quotient map, let R/R be endowed with the quotient topology, and let U be an open set in R/R. Prove that if n ∈ Z is such that p(n) ∈ U then p(n − 1) ∈ U .

(b) Deduce that the open sets in R/R are ∅, R/R and the image sets p(−∞, n] , where n ∈ Z.

(c) Consider the map I : R → Z, x ↦→ I(x) . Is the map I continuous (when Z is endowed with the induced topology) ?

Prove that I deﬁnes a bijection I : R/R → Z. What is the topology on Z making I a homeomorphism ?

I can see that p(n) ∈ U, let U =(x,y), then n ∈ (x,y), p(n-1) = n-1 ∈(x-1,y-1), p^-1(U) ∈(x-1, y), but I'm not sure if it's correct and how to proceed with the rest of the questions. Any input is appreciated!