Just a little bit confused over the notion of homotopy of paths relative to a subset, let's say I have two paths f and g from [0,1] to a space X, and they're homotopic relative to a subspace A of X. If I were able to graph these out, am I correct in saying that they can meander away from each other as they go, but as they pass through A, they are essentially the same line, i.e. one is laid over the other?
And then for the question of fundamental groups, if I have two loops in the same equivalence class, are they basically just one laid on top of the other? Or can there be a situation where two loops take different paths but are in the same class?