Should pathfinder in A* hold closedSet and openedSet or each object should hold its sets?

I am about to implement A* pathfinding algorithm and I wonder how should I implement this - from the point of view of architecture.

I have the pathfinder as a class - I think I will instantiate only one object of this class (or maybe make it a Singleton - this is not so important).

The hardest part for me is whether the `closedSet` and `openedSet` should be attached to objects that can find the path for them or should be stored in `pathfinder` class ?

I am opened to any hints and critique whatsoever.

What is the best practice considering pathfinding in terms of design ?