If denotes the collection of open sets in under the Euclidean metric

and denotes the collection of open sets in under the discrete metric

Show , but

I am pretty sure the only open set in under the discrete metric is . Open balls with radius less than one are sets with a single element, i.e. not open. Open balls with radius greater than 1 are the whole set.

can be expressed as a union of open balls. None of which would be in .

But would these sets be in ? If they are, how is ? Are and sets of sets? Is = or is = { } (that is the set containing the set as its only element)

Could someone clear up this confusion for me? Thank you.