EDB — 225

[225]Difficulty:*.

Let \(Y\) be a topological space. We say that \(Y\) satisfies the property (P) with respect to a topological space \(X\) when it satisfies this condition: for every dense subset \(A⊆ X\) and every pair of continuous functions \(f,g: X→ Y\) such that \(f(a)=g(a)\) for every \(a∈ A\), necessarily there follows that \(f=g\).

Prove that \(Y\) is Hausdorff if and only if it satisfies the property (P) with respect to any topological space \(X\).