Is a closed $G_delta$ set in a Hausdorff space always a zero set?
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I have been trying to prove that if $A$ is a closed set which is also an intersection of countably many open sets then $A$ is the zero set for some continuous real-valued function however have thus far failed. Is this even true?
general-topology functional-analysis
edited Feb 2 '13 at 21:36
Martin
6,9022145
asked Feb 2 '13 at 18:00
Thomas MartinThomas Martin
311
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add a comment |
$begingroup$
I have been trying to prove that if $A$ is a closed set which is also an intersection of countably many open sets then $A$ is the zero set for some continuous real-valued function however have thus far failed. Is this even true?
general-topology functional-analysis
edited Feb 2 '13 at 21:36
Martin
6,9022145
asked Feb 2 '13 at 18:00
Thomas MartinThomas Martin
311
$endgroup$
add a comment |
$begingroup$
I have been trying to prove that if $A$ is a closed set which is also an intersection of countably many open sets then $A$ is the zero set for some continuous real-valued function however have thus far failed. Is this even true?
general-topology functional-analysis
edited Feb 2 '13 at 21:36
Martin
6,9022145
asked Feb 2 '13 at 18:00
Thomas MartinThomas Martin
311
$endgroup$
I have been trying to prove that if $A$ is a closed set which is also an intersection of countably many open sets then $A$ is the zero set for some continuous real-valued function however have thus far failed. Is this even true?
general-topology functional-analysis
general-topology functional-analysis
edited Feb 2 '13 at 21:36
Martin
6,9022145
asked Feb 2 '13 at 18:00
Thomas MartinThomas Martin
311
edited Feb 2 '13 at 21:36
Martin
6,9022145
asked Feb 2 '13 at 18:00
Thomas MartinThomas Martin
311
edited Feb 2 '13 at 21:36
Martin
6,9022145
edited Feb 2 '13 at 21:36
Martin
6,9022145
edited Feb 2 '13 at 21:36
Martin
6,9022145
6,9022145
asked Feb 2 '13 at 18:00
Thomas MartinThomas Martin
311
asked Feb 2 '13 at 18:00
Thomas MartinThomas Martin
311
asked Feb 2 '13 at 18:00
Thomas MartinThomas Martin
311
311
add a comment |
add a comment |
3 Answers
3
active
oldest
votes
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It is not true.
John Thomas, A regular space, not completely regular, Amer. Math. Monthly 76 (1969), 181-182, constructed a regular Hausdorff space $X$ with two points, $p$, and $q$, such that for each continuous $f:XtoBbb R$, $f(a)=f(b)$. Moreover, $X$ has countable local bases at $p$ and $q$. Thus, ${p}$ is a closed $G_delta$-set in $X$ that cannot be a zero-set: any zero-set containing $p$ must also contain $q$.
A. Mysior, A regular space which is not completely regular, Proc. Amer. Math. Soc. 81 (1981), 652-653, is freely available and has a regular Hausdorff space with a point $p$ and a closed set $F$ such that for each continuous $f:XtoBbb R$ and $xin F$, $f(x)=f(p)$. The point $p$ has a countable local base, so here again ${p}$ is a closed $G_delta$ that cannot be a zero-set.
answered Feb 2 '13 at 18:55
Brian M. ScottBrian M. Scott
457k38510911
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add a comment |
$begingroup$
Not even in completely regular Hausdorff spaces. In general we have
$$
text{compact $G_delta$}qquadLongrightarrowqquad
text{zero-set}qquadLongrightarrowqquad
text{closed $G_delta$}
$$
but none reversible.
answered Feb 2 '13 at 22:52
GEdgarGEdgar
62.5k267171
$endgroup$
$begingroup$
A bit late too the party, but could you point me to a reference to the proof of compact $G_delta implies $ zero-set?
$endgroup$
– John
Aug 29 '17 at 11:45
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what can be example for completely regular space?
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– Sushil
Mar 9 '18 at 18:49
add a comment |
$begingroup$
Brian's answer covers the question fully. For fun, here's another example:
Bing's irrational slope space is a countable and connected Hausdorff space.
Now observe:
If $f colon X to mathbb{R}$ is continuous, then $f(X)$ is a countable and connected subset of $mathbb{R}$, hence it must be reduced to a point. Therefore all continuous functions $f colon X to mathbb{R}$ are constant, and the only zero sets are the empty set and the space itself.
Since $X$ is countable and $T_1$, every subset $F$ of $X$ is a $G_delta$-set: $F = bigcap_{x in X setminus F} X setminus {x}$, in particular, there is an abundance of closed $G_delta$ sets that are not zero sets.
answered Feb 2 '13 at 22:19
MartinMartin
6,9022145
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add a comment |
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3 Answers
3
active
oldest
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3 Answers
3
active
oldest
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active
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votes
$begingroup$
It is not true.
John Thomas, A regular space, not completely regular, Amer. Math. Monthly 76 (1969), 181-182, constructed a regular Hausdorff space $X$ with two points, $p$, and $q$, such that for each continuous $f:XtoBbb R$, $f(a)=f(b)$. Moreover, $X$ has countable local bases at $p$ and $q$. Thus, ${p}$ is a closed $G_delta$-set in $X$ that cannot be a zero-set: any zero-set containing $p$ must also contain $q$.
A. Mysior, A regular space which is not completely regular, Proc. Amer. Math. Soc. 81 (1981), 652-653, is freely available and has a regular Hausdorff space with a point $p$ and a closed set $F$ such that for each continuous $f:XtoBbb R$ and $xin F$, $f(x)=f(p)$. The point $p$ has a countable local base, so here again ${p}$ is a closed $G_delta$ that cannot be a zero-set.
answered Feb 2 '13 at 18:55
Brian M. ScottBrian M. Scott
457k38510911
$endgroup$
add a comment |
$begingroup$
It is not true.
John Thomas, A regular space, not completely regular, Amer. Math. Monthly 76 (1969), 181-182, constructed a regular Hausdorff space $X$ with two points, $p$, and $q$, such that for each continuous $f:XtoBbb R$, $f(a)=f(b)$. Moreover, $X$ has countable local bases at $p$ and $q$. Thus, ${p}$ is a closed $G_delta$-set in $X$ that cannot be a zero-set: any zero-set containing $p$ must also contain $q$.
A. Mysior, A regular space which is not completely regular, Proc. Amer. Math. Soc. 81 (1981), 652-653, is freely available and has a regular Hausdorff space with a point $p$ and a closed set $F$ such that for each continuous $f:XtoBbb R$ and $xin F$, $f(x)=f(p)$. The point $p$ has a countable local base, so here again ${p}$ is a closed $G_delta$ that cannot be a zero-set.
answered Feb 2 '13 at 18:55
Brian M. ScottBrian M. Scott
457k38510911
$endgroup$
add a comment |
$begingroup$
It is not true.
John Thomas, A regular space, not completely regular, Amer. Math. Monthly 76 (1969), 181-182, constructed a regular Hausdorff space $X$ with two points, $p$, and $q$, such that for each continuous $f:XtoBbb R$, $f(a)=f(b)$. Moreover, $X$ has countable local bases at $p$ and $q$. Thus, ${p}$ is a closed $G_delta$-set in $X$ that cannot be a zero-set: any zero-set containing $p$ must also contain $q$.
A. Mysior, A regular space which is not completely regular, Proc. Amer. Math. Soc. 81 (1981), 652-653, is freely available and has a regular Hausdorff space with a point $p$ and a closed set $F$ such that for each continuous $f:XtoBbb R$ and $xin F$, $f(x)=f(p)$. The point $p$ has a countable local base, so here again ${p}$ is a closed $G_delta$ that cannot be a zero-set.
answered Feb 2 '13 at 18:55
Brian M. ScottBrian M. Scott
457k38510911
$endgroup$
It is not true.
John Thomas, A regular space, not completely regular, Amer. Math. Monthly 76 (1969), 181-182, constructed a regular Hausdorff space $X$ with two points, $p$, and $q$, such that for each continuous $f:XtoBbb R$, $f(a)=f(b)$. Moreover, $X$ has countable local bases at $p$ and $q$. Thus, ${p}$ is a closed $G_delta$-set in $X$ that cannot be a zero-set: any zero-set containing $p$ must also contain $q$.
A. Mysior, A regular space which is not completely regular, Proc. Amer. Math. Soc. 81 (1981), 652-653, is freely available and has a regular Hausdorff space with a point $p$ and a closed set $F$ such that for each continuous $f:XtoBbb R$ and $xin F$, $f(x)=f(p)$. The point $p$ has a countable local base, so here again ${p}$ is a closed $G_delta$ that cannot be a zero-set.
answered Feb 2 '13 at 18:55
Brian M. ScottBrian M. Scott
457k38510911
answered Feb 2 '13 at 18:55
Brian M. ScottBrian M. Scott
457k38510911
answered Feb 2 '13 at 18:55
Brian M. ScottBrian M. Scott
457k38510911
answered Feb 2 '13 at 18:55
Brian M. ScottBrian M. Scott
457k38510911
457k38510911
add a comment |
add a comment |
$begingroup$
Not even in completely regular Hausdorff spaces. In general we have
$$
text{compact $G_delta$}qquadLongrightarrowqquad
text{zero-set}qquadLongrightarrowqquad
text{closed $G_delta$}
$$
but none reversible.
answered Feb 2 '13 at 22:52
GEdgarGEdgar
62.5k267171
$endgroup$
$begingroup$
A bit late too the party, but could you point me to a reference to the proof of compact $G_delta implies $ zero-set?
$endgroup$
– John
Aug 29 '17 at 11:45
$begingroup$
what can be example for completely regular space?
$endgroup$
– Sushil
Mar 9 '18 at 18:49
add a comment |
$begingroup$
Not even in completely regular Hausdorff spaces. In general we have
$$
text{compact $G_delta$}qquadLongrightarrowqquad
text{zero-set}qquadLongrightarrowqquad
text{closed $G_delta$}
$$
but none reversible.
answered Feb 2 '13 at 22:52
GEdgarGEdgar
62.5k267171
$endgroup$
$begingroup$
A bit late too the party, but could you point me to a reference to the proof of compact $G_delta implies $ zero-set?
$endgroup$
– John
Aug 29 '17 at 11:45
$begingroup$
what can be example for completely regular space?
$endgroup$
– Sushil
Mar 9 '18 at 18:49
add a comment |
$begingroup$
Not even in completely regular Hausdorff spaces. In general we have
$$
text{compact $G_delta$}qquadLongrightarrowqquad
text{zero-set}qquadLongrightarrowqquad
text{closed $G_delta$}
$$
but none reversible.
answered Feb 2 '13 at 22:52
GEdgarGEdgar
62.5k267171
$endgroup$
Not even in completely regular Hausdorff spaces. In general we have
$$
text{compact $G_delta$}qquadLongrightarrowqquad
text{zero-set}qquadLongrightarrowqquad
text{closed $G_delta$}
$$
but none reversible.
answered Feb 2 '13 at 22:52
GEdgarGEdgar
62.5k267171
answered Feb 2 '13 at 22:52
GEdgarGEdgar
62.5k267171
answered Feb 2 '13 at 22:52
GEdgarGEdgar
62.5k267171
answered Feb 2 '13 at 22:52
GEdgarGEdgar
62.5k267171
62.5k267171
$begingroup$
A bit late too the party, but could you point me to a reference to the proof of compact $G_delta implies $ zero-set?
$endgroup$
– John
Aug 29 '17 at 11:45
$begingroup$
what can be example for completely regular space?
$endgroup$
– Sushil
Mar 9 '18 at 18:49
add a comment |
$begingroup$
A bit late too the party, but could you point me to a reference to the proof of compact $G_delta implies $ zero-set?
$endgroup$
– John
Aug 29 '17 at 11:45
$begingroup$
what can be example for completely regular space?
$endgroup$
– Sushil
Mar 9 '18 at 18:49
$begingroup$
A bit late too the party, but could you point me to a reference to the proof of compact $G_delta implies $ zero-set?
$endgroup$
– John
Aug 29 '17 at 11:45
$begingroup$
A bit late too the party, but could you point me to a reference to the proof of compact $G_delta implies $ zero-set?
$endgroup$
– John
Aug 29 '17 at 11:45
$begingroup$
what can be example for completely regular space?
$endgroup$
– Sushil
Mar 9 '18 at 18:49
$begingroup$
what can be example for completely regular space?
$endgroup$
– Sushil
Mar 9 '18 at 18:49
add a comment |
$begingroup$
Brian's answer covers the question fully. For fun, here's another example:
Bing's irrational slope space is a countable and connected Hausdorff space.
Now observe:
If $f colon X to mathbb{R}$ is continuous, then $f(X)$ is a countable and connected subset of $mathbb{R}$, hence it must be reduced to a point. Therefore all continuous functions $f colon X to mathbb{R}$ are constant, and the only zero sets are the empty set and the space itself.
Since $X$ is countable and $T_1$, every subset $F$ of $X$ is a $G_delta$-set: $F = bigcap_{x in X setminus F} X setminus {x}$, in particular, there is an abundance of closed $G_delta$ sets that are not zero sets.
answered Feb 2 '13 at 22:19
MartinMartin
6,9022145
$endgroup$
add a comment |
$begingroup$
Brian's answer covers the question fully. For fun, here's another example:
Bing's irrational slope space is a countable and connected Hausdorff space.
Now observe:
If $f colon X to mathbb{R}$ is continuous, then $f(X)$ is a countable and connected subset of $mathbb{R}$, hence it must be reduced to a point. Therefore all continuous functions $f colon X to mathbb{R}$ are constant, and the only zero sets are the empty set and the space itself.
Since $X$ is countable and $T_1$, every subset $F$ of $X$ is a $G_delta$-set: $F = bigcap_{x in X setminus F} X setminus {x}$, in particular, there is an abundance of closed $G_delta$ sets that are not zero sets.
answered Feb 2 '13 at 22:19
MartinMartin
6,9022145
$endgroup$
add a comment |
$begingroup$
Brian's answer covers the question fully. For fun, here's another example:
Bing's irrational slope space is a countable and connected Hausdorff space.
Now observe:
If $f colon X to mathbb{R}$ is continuous, then $f(X)$ is a countable and connected subset of $mathbb{R}$, hence it must be reduced to a point. Therefore all continuous functions $f colon X to mathbb{R}$ are constant, and the only zero sets are the empty set and the space itself.
Since $X$ is countable and $T_1$, every subset $F$ of $X$ is a $G_delta$-set: $F = bigcap_{x in X setminus F} X setminus {x}$, in particular, there is an abundance of closed $G_delta$ sets that are not zero sets.
answered Feb 2 '13 at 22:19
MartinMartin
6,9022145
$endgroup$
Brian's answer covers the question fully. For fun, here's another example:
Bing's irrational slope space is a countable and connected Hausdorff space.
Now observe:
If $f colon X to mathbb{R}$ is continuous, then $f(X)$ is a countable and connected subset of $mathbb{R}$, hence it must be reduced to a point. Therefore all continuous functions $f colon X to mathbb{R}$ are constant, and the only zero sets are the empty set and the space itself.
Since $X$ is countable and $T_1$, every subset $F$ of $X$ is a $G_delta$-set: $F = bigcap_{x in X setminus F} X setminus {x}$, in particular, there is an abundance of closed $G_delta$ sets that are not zero sets.
answered Feb 2 '13 at 22:19
MartinMartin
6,9022145
edited Feb 2 '13 at 22:36
answered Feb 2 '13 at 22:19
MartinMartin
6,9022145
answered Feb 2 '13 at 22:19
MartinMartin
6,9022145
answered Feb 2 '13 at 22:19
MartinMartin
6,9022145
6,9022145
add a comment |
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