Question

prove that for any two distinct points a, b in a metric space ) (x, d there exist disjoint open spheres with centres a and b respectively..​

Answer 1

Step-by-step explanation:

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Answer 2

Answer:

assume that you know that open balls in a metric space are open sets. Then since x1 and x2 are distinct

r=12d(x1,x2)>0.

Then the balls Br(x1) and Br(x2) are disjoint open sets containing x1 and x2 respectively. If there were a point p in both balls then

d(x1,x2)≤d(x1,p)+d(p,x2)<r+r=d(x1,x2).

Clearly this is impossible, so the sets are disjoint.