- #1
rumjum
- 40
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Homework Statement
Suppose (X,d) is a metric space. For a point in X and a non empty set S (as a subset of X), define d(p,S) = inf({(d(p,x):x belongs to S}). Prove that the closure of S is equal to the set {p belongs to S : d(p,S) =0}
Homework Equations
Closure of S = S U S' , where S' is the set of limit points of S.
The Attempt at a Solution
Any hint would be useful. My line of reasoning is the following.
I started with a picture of all the metric space. I took cases like p being part of S and then p not being part of S. But, if p is not part of S, then the intersection of Neighborhood of p with some small radius , r with the set S could be null as d(p,x) > 0. Where as the least distance of any such point "p" and x is zero. Also, d(p,S) belongs to R. Hence, there shall be a real number between any nonzero d(p,S) and zero. As a result, all elements of inf{d(p,S)} = 0.