1. Consider the metric space X = (Q ∩ [0; 3]; dE):
(a) the the point 2 is an interior point of the subset A of X where
A = {x ∈ Q | 1 ≤ x ≤ 3}?
True. Since you can make an open ball around 2.
(b) The the point 2 is an interior point of the subset B of X where
B = {x ∈ Q | 2 ≤ x ≤ 3}?
False. Since you can't make an open ball around 2 that is contained in the set.
(c) The point 3 is an interior point of the subset C of X where
C = {x ∈ Q | 2 < x ≤ 3}?
True. Since you can construct a ball around 3, where all the points in the ball is in the metric space.
(d) Describe the possible forms that an open ball can take in X = (Q ∩ [0; 3]; dE).
I don't really get this question, but I think the possible forms are all balls in [0,3] . IS THAT RIGHT???
Is the other answers right.
yes that is the space.
I was wondering with reference to Q1. is that whether you can make an open ball around 2, since that ball will contain irrational numbers which are not in the set,since we are working in the world of the rationals, or does this even matter.