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I am reading an article and there is something I find odd. The setting is a Banach space E and we have two disjoint closed subsets A and B of E. There is no additional assumption on E, A or B. The author then says,

"Let f:E-->[0,1] be a Urysohn's function such that f(x)=0 if and only if x is in A, and f(x)=1 on B."

But never have I seen a version of Urysohn's lemma that guarantees that f(x)=0 ifand only ifx is in A.

Does someone have an explanation? (I would ask my advisor but she had gone on vacation for 3 weeks)

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# Stronger Urysohn lemma?

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