The following is from Mariano's comments on my earlier question
In a topological vector space, why is the following true:
if a neighborhood U of zero contains a scaled copy of the whole space, then it is in fact the whole space.
Is "a neighborhood U of zero contains a scaled copy of the whole space" the same as "a scaled copy of a neighborhood U of zero is the whole space"?
I have thought about this for a while but don't know why.
In a vector space, is it true that if a subset U of zero contains a scaled copy of the whole space, then it is in fact the whole space? I think it is not true when the base field of the vector space is a finite set?
Is "a subset U of zero contains a scaled copy of the whole space" the same as "a scaled copy of a subset U of zero is the whole space"?
Thanks and regards!