how is the topology defined on Y? is d' the same metic as d? what is the subset F'?

let me give you an example of how a relative topology works, and why the open sets are different than in the larger space.

suppose X = R, the real numbers, and Y = [0,1], the unit interval. then the set (1/2,1] is not open in X, but it IS open in Y.

you see, if the definition of an open set was the same in Y as it was in X, the element 1 in Y would have NO neighborhoods at all!

and, by the very definition of a topology, the entire space is an open set. so Y = [0,1] is open, but [0,1] is certainly not open in X.

so the definition of a relative topology on a subspace Y of X is σ = {A∩Y: A is open in X}. if your subspace Y has the same metric, d,

then you should have either the statement about the N' or the statement about the O' as given.