Let X d be a metric pace and E X a closed subset For a point

Let (X, d) be a metric pace and E X a closed subset. For a point x X define its distance d(x, E) from the set E by d(x, E) = inf{d(x, y) : y E}. Show that d(x, E) = 0 if and only if x E. Show that for every closed subset E X there is a continuous function f : X rightarrow R such that f^-1(0) is exactly E.

Solution

a) Given that (x,d) is a metric space

E is a closed subset

As elements of E also belong to X and X is a metric space

d(x,E) = 0 implies x belongs to E

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b) Let us assume that if possible f inverse of 0 is not in E

Then 0 is not in X

This implies a contradiction for metric space

Hence f inverse of 0 is in E