For every closed set $F$ in a completely regular space $X,$ does there exist a nonzero continuous function...
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Let $X$ be a completely regular space, that is, for every closed set $Fsubseteq X$ and $xnotin F,$ there exists a continuous function $g:Xto [0,1]$ such that $g(F) = {0}$ and $g(x) =1.$
Question: For every closed set $Fsubseteq X,$ does there exist a nonzero continuous function $f:Xto [0,1]$ such that $f=0$ outside $F?$
general-topology separation-axioms
asked Jan 5 at 12:01
IdonknowIdonknow
2,603950119
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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.
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$begingroup$
Let $X$ be a completely regular space, that is, for every closed set $Fsubseteq X$ and $xnotin F,$ there exists a continuous function $g:Xto [0,1]$ such that $g(F) = {0}$ and $g(x) =1.$
Question: For every closed set $Fsubseteq X,$ does there exist a nonzero continuous function $f:Xto [0,1]$ such that $f=0$ outside $F?$
general-topology separation-axioms
asked Jan 5 at 12:01
IdonknowIdonknow
2,603950119
$endgroup$
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.
add a comment |
$begingroup$
Let $X$ be a completely regular space, that is, for every closed set $Fsubseteq X$ and $xnotin F,$ there exists a continuous function $g:Xto [0,1]$ such that $g(F) = {0}$ and $g(x) =1.$
Question: For every closed set $Fsubseteq X,$ does there exist a nonzero continuous function $f:Xto [0,1]$ such that $f=0$ outside $F?$
general-topology separation-axioms
asked Jan 5 at 12:01
IdonknowIdonknow
2,603950119
$endgroup$
Let $X$ be a completely regular space, that is, for every closed set $Fsubseteq X$ and $xnotin F,$ there exists a continuous function $g:Xto [0,1]$ such that $g(F) = {0}$ and $g(x) =1.$
Question: For every closed set $Fsubseteq X,$ does there exist a nonzero continuous function $f:Xto [0,1]$ such that $f=0$ outside $F?$
general-topology separation-axioms
general-topology separation-axioms
asked Jan 5 at 12:01
IdonknowIdonknow
2,603950119
asked Jan 5 at 12:01
IdonknowIdonknow
2,603950119
asked Jan 5 at 12:01
IdonknowIdonknow
2,603950119
asked Jan 5 at 12:01
IdonknowIdonknow
2,603950119
asked Jan 5 at 12:01
IdonknowIdonknow
2,603950119
2,603950119
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.
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.
add a comment |
add a comment |
1 Answer
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Not true even in the real line. If $F$ is a singleton set then any continuous function vanishing outside ${x}$ is identically $0$.
edited Jan 5 at 12:10
Henno Brandsma
115k349125
answered Jan 5 at 12:06
Kavi Rama MurthyKavi Rama Murthy
72.9k53170
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add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Not true even in the real line. If $F$ is a singleton set then any continuous function vanishing outside ${x}$ is identically $0$.
edited Jan 5 at 12:10
Henno Brandsma
115k349125
answered Jan 5 at 12:06
Kavi Rama MurthyKavi Rama Murthy
72.9k53170
$endgroup$
add a comment |
$begingroup$
Not true even in the real line. If $F$ is a singleton set then any continuous function vanishing outside ${x}$ is identically $0$.
edited Jan 5 at 12:10
Henno Brandsma
115k349125
answered Jan 5 at 12:06
Kavi Rama MurthyKavi Rama Murthy
72.9k53170
$endgroup$
add a comment |
$begingroup$
Not true even in the real line. If $F$ is a singleton set then any continuous function vanishing outside ${x}$ is identically $0$.
edited Jan 5 at 12:10
Henno Brandsma
115k349125
answered Jan 5 at 12:06
Kavi Rama MurthyKavi Rama Murthy
72.9k53170
$endgroup$
Not true even in the real line. If $F$ is a singleton set then any continuous function vanishing outside ${x}$ is identically $0$.
edited Jan 5 at 12:10
Henno Brandsma
115k349125
answered Jan 5 at 12:06
Kavi Rama MurthyKavi Rama Murthy
72.9k53170
edited Jan 5 at 12:10
Henno Brandsma
115k349125
edited Jan 5 at 12:10
Henno Brandsma
115k349125
edited Jan 5 at 12:10
Henno Brandsma
115k349125
115k349125
answered Jan 5 at 12:06
Kavi Rama MurthyKavi Rama Murthy
72.9k53170
answered Jan 5 at 12:06
Kavi Rama MurthyKavi Rama Murthy
72.9k53170
answered Jan 5 at 12:06
Kavi Rama MurthyKavi Rama Murthy
72.9k53170
72.9k53170
add a comment |
add a comment |
