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On the equality $\{f\in C(X), f|_A=0\}+\{f\in C(X), f|_B=0\}=\{f\in C(X), f|_{A\cap B}=0\}$
Let $X$ be a Tychonoff space, let $A,B\subset X$ be closed. Let $J_A$ be the set of all continuous on $X$ real-valued functions which vanish on $A$.
For which $X$'s is it true that $J_A+J_B=J_{A\cap ...
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