In a metric space (S,d), let A be non-empty subset of S. Define
a function f_A(x): S→R by the formula
f_A(x) = inf { d(x,y) : y in A } for each x in S.
The value f_A(x) is called the distance from x to A.
(a) Prove that f_A is uniformly continuous on S.
(b) Prove cl(A) = { x in S : f_A(x) = 0 }, where cl(A) means
the closure of A.
依上, 我們可以有下列的事實:
In a metric space (S,d), let A and B be disjoint closed subsets
of S. Prove that there are two disjoint open sets U and V in S
such that A < U and B < V.
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