1. The problem statement, all variables and given/known data Let X and Y be ordered sets in the order topology. I want to show that a function f:X→Y is injective. We are given that f is surjective and preserves order. 2. Relevant equations Definition of an order preserving map: If x≤y implies f(x)≤f(y) 3. The attempt at a solution So if we assume that it is not injective then we are assuming that f(x)=f(y) and x[itex]\neq[/itex]y. Then either x>y or y<x. Assuming both, would clearly be a contradiction since it can only be one or the other in an ordered set. Now I need to show that x>y raises a contradiction and similarly x<y also raises a contradiction. My main question here is if x<y is it a contradiction if f(x)=f(y)? It seems like if order is preserved then if x<y then f(x)<f(y) but by the definition, it appears that if x<y then f(x)<f(y) or f(x)=f(y). Thank you for your time.