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Suppose a Hausdorff topological space $(X, \tau)$ has the following property. A and B are disjoint closed subsets of $X$ implies there is a continuous function say $f_{AB} : X \rightarrow [0,1]$ such that $f(x)=0$ if $x \in A$ and $f(x)=1$if $x \in B$, then $(X, \tau)$ is a normal space. Can someone help me with this?

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    $\begingroup$ Look at preimages $f^{-1}(1/2,1]$ and $f^{-1}[0,1/2)$. Recall what normality means. $\endgroup$ – Luiz Cordeiro May 2 '16 at 20:23
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HINT: $\left[0,\frac13\right)$ and $\left(\frac23,1\right]$ are disjoint open sets in $[0,1]$.

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