Question

the distance between two non empty subsets Aand B of a metric space Xis defined by

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Answer 1

Answer:

Let (X,d)(X,d) be a metric space. Define the distance between two nonempty subsets AA and BB of XX by

d(A,B)=inf{d(x,y):x∈Aandy∈B}

d(A,B)=inf{d(x,y):x∈Aandy∈B}

(a) Give an example of two closed sets AA and BB with A∩B=∅A∩B=∅ and such that d(A,B)=0.d(A,B)=0.

(b) If A∩B=∅,AA∩B=∅,A is clossed and BB is closed and bounded (both nonempty) then show that d(A,B)>0.d(A,B)>0.

So for the first part I thought if I chose the metric space RR and let A={0}A={0} and B={y∈R:1≤y≤10}B={y∈R:1≤y≤10}

These are two subsets of RR and the are closed and their intersection is empty.And if I understand the distance formula correctly it is the glb of any interval d(x,y)d(x,y) and in my case any interval d(0,y)d(0,y)

But then if this is true I do not really understand what I am supposed to be accomplishing in part (b)

I would appreciate a nudge in the right direction, and confirmation at my attempt of part (a)