1
$\begingroup$

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 $F?$

$\endgroup$

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.

3
$\begingroup$

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

$\endgroup$

Not the answer you're looking for? Browse other questions tagged or ask your own question.