# For every closed set $F$ in a completely regular space $X,$ does there exist a nonzero continuous function $f:X\to [0,1]$ such that $f=0$ outside $F?$ [closed]

Let $$X$$ be a completely regular space, that is, for every closed set $$F\subseteq X$$ and $$x\not\in F,$$ there exists a continuous function $$g:X\to [0,1]$$ such that $$g(F) = \{0\}$$ and $$g(x) =1.$$

Question: For every closed set $$F\subseteq X,$$ does there exist a nonzero continuous function $$f:X\to [0,1]$$ such that $$f=0$$ outside $$F?$$

## closed as off-topic by Nosrati, user91500, José Carlos Santos, Cesareo, Alexander Gruber♦Jan 9 at 2:06

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## 1 Answer

Not true even in the real line. If $$F$$ is a singleton set then any continuous function vanishing outside $$\{x\}$$ is identically $$0$$.