Please help me in finding a possible solution for this problem that doesn't involve the whole space or the empty set .... b/c my professor made that requirement.
Problem.
Give an example of a non-empty space X with metric d, in which every closed set is the intersection of a non-empty family of closed balls.
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okay. i did consider the simple space with only two elements but i am puzzled on the closed balls.
i know that a closed ball is C(a) ={x in X: d(a,x) <=r}
but i don't have any representation of what it actually looks like? can it leave the set? or am i missing the obvious and is the radius = to every point in between and including 1?
The entire space is the two points p and q, so your question of whether it leaves the set makes no sense.
The closed balls of radius less than one consists of one point, and the closed balls or radius at least one consists of both points. This is all the closed balls.
The closed sets are: the empty set, {p}, {q} and {p,q}=X.