4.II.14A
Let be a metric space, and a non-empty closed subset of . For , set
Prove that is a continuous function of , and that it is strictly positive for .
A topological space is called normal if for any pair of disjoint closed subsets , there exist disjoint open subsets . By considering the function
or otherwise, deduce that any metric space is normal.
Suppose now that is a normal topological space, and that are disjoint closed subsets in . Prove that there exist open subsets , whose closures are disjoint. In the case when with the standard metric topology, and , find explicit open subsets with the above property.
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