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.
Oh ok. The single sets are open and are in F.
All unions of the single sets are open under the discrete metric as well.
So I can build any open set in E using unions.
But (x_1,x_2,...,x_n) union (y_1,y_2,...,y_n) could be an open set if F
but it is certainly not in E