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)?