Metric and Topological Spaces | Part IB, 2005

Let (M,d)(M, d) be a metric space, and FF a non-empty closed subset of MM. For xMx \in M, set

d(x,F)=infzFd(x,z)d(x, F)=\inf _{z \in F} d(x, z)

Prove that d(x,F)d(x, F) is a continuous function of xx, and that it is strictly positive for xFx \notin F.

A topological space is called normal if for any pair of disjoint closed subsets F1,F2F_{1}, F_{2}, there exist disjoint open subsets U1F1,U2F2U_{1} \supset F_{1}, U_{2} \supset F_{2}. By considering the function

d(x,F1)d(x,F2)d\left(x, F_{1}\right)-d\left(x, F_{2}\right)

or otherwise, deduce that any metric space is normal.

Suppose now that XX is a normal topological space, and that F1,F2F_{1}, F_{2} are disjoint closed subsets in XX. Prove that there exist open subsets W1F1,W2F2W_{1} \supset F_{1}, W_{2} \supset F_{2}, whose closures are disjoint. In the case when X=R2X=\mathbf{R}^{2} with the standard metric topology, F1={(x,1/x):x<0}F_{1}=\{(x,-1 / x): x<0\} and F2={(x,1/x):x>0}F_{2}=\{(x, 1 / x): x>0\}, find explicit open subsets W1,W2W_{1}, W_{2} with the above property.

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