# 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

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Nosrati, user91500, José Carlos Santos, Cesareo, Alexander Gruber
If this question can be reworded to fit the rules in the help center, please edit the question.

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