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On the equality $\{f\in C(X), f|_A=0\}+\{f\in C(X), f|_B=0\}=\{f\in C(X), f|_{A\cap B}=0\}$

This characterizes normality. That it implies normality was observed above by Remy. Conversely, assume $f$ vanishes on $A\cap B$. Define $h:A\cup B\to\mathbb{R}$ by $h(x)=f(x)$ if $x\in A$ and $h(x)=0$...
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