Here's the idea. Take as a simple example. Then, we can form a decomposition as you call it (I'd call it a partition) by evidently . So, I want to topologize by saying that is open if and only if is open in . For example, is open since which is open. Whereas is not open since is not open in . Make sense?
For the actual exercise, if it makes it easier for you to think about it, define [tex]\sim [tex] on by saying if and only if there exists such that . Then, the quotient map is where is evidently the equivalence class of under .