It's hard for us to guess what exactly "non-trivial topology" is in this context. A topology on a set X is a collection of subsets of X that satisfies a number of properties. A member of the topology is called an open set. These open sets are useful because we can use them to assign a meaning to statements like "this sequence converges to this point" or "this function is continuous".
The two simplest ways to define a topology on a given set X is to choose the topology to be either "all subsets of X" or "just the sets ∅ and X". These choices are considered trivial, and as you can guess, they are pretty useless. For example, what's the point of the concept of continuity if all functions are continuous? I don't think that your "non-trivial topology" has anything to do with this.
If X is a set and T is a topology on X, then the pair (X,T) is called a topological space. If (X,T) and (Y,S) are topological spaces, it may or may not be true that there exists a bijection from X into Y that's continuous with respect to the topologies T and S. If such a function exists, then we describe this situation with phrases like "X and Y are topologically equivalent". (Actually, "homeomorphic" is the correct technical term).
When two spaces are
not topologically equivalent, there's typically something fundamentally different about their shapes. An example that's often brought up is that a donut is topologically equivalent to a coffee mug, but not to a sphere. This is because the former two shapes both have a hole in them, and a sphere doesn't.
My guess is that your non-trivial topology has something to do with this sort of thing. For example, a surface that's "like" a donut rather than "like" a sphere.
