A question about zero-sets
The set $C(X)$ of all continuous, real-value functions no a topological space $X$ will be provided with an algebraic structure and order structure.
zero-set means:
$Z(f) = Z_{X} (f) = { x in X : f(x) = 0 } quad ( f in C(X) )$
$Z(X)= Z[C(X)]={ Z(f) : f in C(X) }$
I know that $Z(X)$ closed under countable intersection and also $Z(X)$ need not be closed under infinite union.
1: In a general space, Do countable or finite union of zero-sets be a
zero-set?Why?
2:Is $Z(X)$ closed under arbitrary intersection?Why?
general-topology topological-groups
asked Dec 26 '18 at 18:12
joe
964
add a comment |
The set $C(X)$ of all continuous, real-value functions no a topological space $X$ will be provided with an algebraic structure and order structure.
zero-set means:
$Z(f) = Z_{X} (f) = { x in X : f(x) = 0 } quad ( f in C(X) )$
$Z(X)= Z[C(X)]={ Z(f) : f in C(X) }$
I know that $Z(X)$ closed under countable intersection and also $Z(X)$ need not be closed under infinite union.
1: In a general space, Do countable or finite union of zero-sets be a
zero-set?Why?
2:Is $Z(X)$ closed under arbitrary intersection?Why?
general-topology topological-groups
asked Dec 26 '18 at 18:12
joe
964
2
What is $Z(X)$? What does it mean for $Z(X)$ to be, for instance, "closed under countable intersection"?
– Ben W
Dec 26 '18 at 18:15
@BenW $Z(X)$ is the set of all zero-sets of $X$.
– Henno Brandsma
Dec 26 '18 at 18:22
Hint: Find an example of a closed set that is not a zero set.
– GEdgar
Dec 26 '18 at 18:27
And what does it mean for $Z(X)$ to be closed under countable intersection?
– Ben W
Dec 26 '18 at 18:28
add a comment |
The set $C(X)$ of all continuous, real-value functions no a topological space $X$ will be provided with an algebraic structure and order structure.
zero-set means:
$Z(f) = Z_{X} (f) = { x in X : f(x) = 0 } quad ( f in C(X) )$
$Z(X)= Z[C(X)]={ Z(f) : f in C(X) }$
I know that $Z(X)$ closed under countable intersection and also $Z(X)$ need not be closed under infinite union.
1: In a general space, Do countable or finite union of zero-sets be a
zero-set?Why?
2:Is $Z(X)$ closed under arbitrary intersection?Why?
general-topology topological-groups
asked Dec 26 '18 at 18:12
joe
964
The set $C(X)$ of all continuous, real-value functions no a topological space $X$ will be provided with an algebraic structure and order structure.
zero-set means:
$Z(f) = Z_{X} (f) = { x in X : f(x) = 0 } quad ( f in C(X) )$
$Z(X)= Z[C(X)]={ Z(f) : f in C(X) }$
I know that $Z(X)$ closed under countable intersection and also $Z(X)$ need not be closed under infinite union.
1: In a general space, Do countable or finite union of zero-sets be a
zero-set?Why?
2:Is $Z(X)$ closed under arbitrary intersection?Why?
general-topology topological-groups
general-topology topological-groups
asked Dec 26 '18 at 18:12
joe
964
asked Dec 26 '18 at 18:12
joe
964
edited Dec 26 '18 at 18:24
asked Dec 26 '18 at 18:12
joe
964
asked Dec 26 '18 at 18:12
joe
964
asked Dec 26 '18 at 18:12
joe
964
964
2
What is $Z(X)$? What does it mean for $Z(X)$ to be, for instance, "closed under countable intersection"?
– Ben W
Dec 26 '18 at 18:15
@BenW $Z(X)$ is the set of all zero-sets of $X$.
– Henno Brandsma
Dec 26 '18 at 18:22
Hint: Find an example of a closed set that is not a zero set.
– GEdgar
Dec 26 '18 at 18:27
And what does it mean for $Z(X)$ to be closed under countable intersection?
– Ben W
Dec 26 '18 at 18:28
add a comment |
2
What is $Z(X)$? What does it mean for $Z(X)$ to be, for instance, "closed under countable intersection"?
– Ben W
Dec 26 '18 at 18:15
@BenW $Z(X)$ is the set of all zero-sets of $X$.
– Henno Brandsma
Dec 26 '18 at 18:22
Hint: Find an example of a closed set that is not a zero set.
– GEdgar
Dec 26 '18 at 18:27
And what does it mean for $Z(X)$ to be closed under countable intersection?
– Ben W
Dec 26 '18 at 18:28
2
2
What is $Z(X)$? What does it mean for $Z(X)$ to be, for instance, "closed under countable intersection"?
– Ben W
Dec 26 '18 at 18:15
What is $Z(X)$? What does it mean for $Z(X)$ to be, for instance, "closed under countable intersection"?
– Ben W
Dec 26 '18 at 18:15
@BenW $Z(X)$ is the set of all zero-sets of $X$.
– Henno Brandsma
Dec 26 '18 at 18:22
@BenW $Z(X)$ is the set of all zero-sets of $X$.
– Henno Brandsma
Dec 26 '18 at 18:22
Hint: Find an example of a closed set that is not a zero set.
– GEdgar
Dec 26 '18 at 18:27
Hint: Find an example of a closed set that is not a zero set.
– GEdgar
Dec 26 '18 at 18:27
And what does it mean for $Z(X)$ to be closed under countable intersection?
– Ben W
Dec 26 '18 at 18:28
And what does it mean for $Z(X)$ to be closed under countable intersection?
– Ben W
Dec 26 '18 at 18:28
add a comment |
1 Answer
1
active
oldest
votes
$Z(f)$ is closed under finite unions : $Z(f) cup Z(g) = Z(fg)$ plus induction.
In general not closed under countable unions: singletons are in $Z(mathbb{R})$ but $mathbb{Q}$ is not.
Also $Z(X)$ will not be closed under arbitary intersections, but you need some non-metrisable counterexample (in metric spaces $Z(X)$ is just the collection of all closed sets which is closed under all intersections). Think (or search) about a counterexample with some closed set that is not a zero-set.
answered Dec 26 '18 at 18:28
Henno Brandsma
105k346113
What dose it mean? singletons are in $Z(mathbb{R})$ but $ mathbb{Q}$ is not.
– joe
Dec 26 '18 at 18:38
@joe the rationals are a countable union of zerosets that’s not a zeroset
– Henno Brandsma
Dec 26 '18 at 18:40
In metric space every closed set is zero-set, buy you say "search about a counterexample with some closed set that is not a zero-set". I can not see.
– joe
Dec 26 '18 at 18:43
2
@joe so find a non metric $X$ as an example.
– Henno Brandsma
Dec 26 '18 at 19:00
2
@joe you can use that in a Tychonoff space all closed sets are intersections of zero-sets. Also, a zero-set is a $G_delta$ and the reverse holds in normal spaces.
– Henno Brandsma
Dec 26 '18 at 19:04
add a comment |
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1 Answer
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1 Answer
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active
oldest
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oldest
votes
$Z(f)$ is closed under finite unions : $Z(f) cup Z(g) = Z(fg)$ plus induction.
In general not closed under countable unions: singletons are in $Z(mathbb{R})$ but $mathbb{Q}$ is not.
Also $Z(X)$ will not be closed under arbitary intersections, but you need some non-metrisable counterexample (in metric spaces $Z(X)$ is just the collection of all closed sets which is closed under all intersections). Think (or search) about a counterexample with some closed set that is not a zero-set.
answered Dec 26 '18 at 18:28
Henno Brandsma
105k346113
What dose it mean? singletons are in $Z(mathbb{R})$ but $ mathbb{Q}$ is not.
– joe
Dec 26 '18 at 18:38
@joe the rationals are a countable union of zerosets that’s not a zeroset
– Henno Brandsma
Dec 26 '18 at 18:40
In metric space every closed set is zero-set, buy you say "search about a counterexample with some closed set that is not a zero-set". I can not see.
– joe
Dec 26 '18 at 18:43
2
@joe so find a non metric $X$ as an example.
– Henno Brandsma
Dec 26 '18 at 19:00
2
@joe you can use that in a Tychonoff space all closed sets are intersections of zero-sets. Also, a zero-set is a $G_delta$ and the reverse holds in normal spaces.
– Henno Brandsma
Dec 26 '18 at 19:04
add a comment |
$Z(f)$ is closed under finite unions : $Z(f) cup Z(g) = Z(fg)$ plus induction.
In general not closed under countable unions: singletons are in $Z(mathbb{R})$ but $mathbb{Q}$ is not.
Also $Z(X)$ will not be closed under arbitary intersections, but you need some non-metrisable counterexample (in metric spaces $Z(X)$ is just the collection of all closed sets which is closed under all intersections). Think (or search) about a counterexample with some closed set that is not a zero-set.
answered Dec 26 '18 at 18:28
Henno Brandsma
105k346113
What dose it mean? singletons are in $Z(mathbb{R})$ but $ mathbb{Q}$ is not.
– joe
Dec 26 '18 at 18:38
@joe the rationals are a countable union of zerosets that’s not a zeroset
– Henno Brandsma
Dec 26 '18 at 18:40
In metric space every closed set is zero-set, buy you say "search about a counterexample with some closed set that is not a zero-set". I can not see.
– joe
Dec 26 '18 at 18:43
2
@joe so find a non metric $X$ as an example.
– Henno Brandsma
Dec 26 '18 at 19:00
2
@joe you can use that in a Tychonoff space all closed sets are intersections of zero-sets. Also, a zero-set is a $G_delta$ and the reverse holds in normal spaces.
– Henno Brandsma
Dec 26 '18 at 19:04
add a comment |
$Z(f)$ is closed under finite unions : $Z(f) cup Z(g) = Z(fg)$ plus induction.
In general not closed under countable unions: singletons are in $Z(mathbb{R})$ but $mathbb{Q}$ is not.
Also $Z(X)$ will not be closed under arbitary intersections, but you need some non-metrisable counterexample (in metric spaces $Z(X)$ is just the collection of all closed sets which is closed under all intersections). Think (or search) about a counterexample with some closed set that is not a zero-set.
answered Dec 26 '18 at 18:28
Henno Brandsma
105k346113
$Z(f)$ is closed under finite unions : $Z(f) cup Z(g) = Z(fg)$ plus induction.
In general not closed under countable unions: singletons are in $Z(mathbb{R})$ but $mathbb{Q}$ is not.
Also $Z(X)$ will not be closed under arbitary intersections, but you need some non-metrisable counterexample (in metric spaces $Z(X)$ is just the collection of all closed sets which is closed under all intersections). Think (or search) about a counterexample with some closed set that is not a zero-set.
answered Dec 26 '18 at 18:28
Henno Brandsma
105k346113
answered Dec 26 '18 at 18:28
Henno Brandsma
105k346113
answered Dec 26 '18 at 18:28
Henno Brandsma
105k346113
answered Dec 26 '18 at 18:28
Henno Brandsma
105k346113
105k346113
What dose it mean? singletons are in $Z(mathbb{R})$ but $ mathbb{Q}$ is not.
– joe
Dec 26 '18 at 18:38
@joe the rationals are a countable union of zerosets that’s not a zeroset
– Henno Brandsma
Dec 26 '18 at 18:40
In metric space every closed set is zero-set, buy you say "search about a counterexample with some closed set that is not a zero-set". I can not see.
– joe
Dec 26 '18 at 18:43
2
@joe so find a non metric $X$ as an example.
– Henno Brandsma
Dec 26 '18 at 19:00
2
@joe you can use that in a Tychonoff space all closed sets are intersections of zero-sets. Also, a zero-set is a $G_delta$ and the reverse holds in normal spaces.
– Henno Brandsma
Dec 26 '18 at 19:04
add a comment |
What dose it mean? singletons are in $Z(mathbb{R})$ but $ mathbb{Q}$ is not.
– joe
Dec 26 '18 at 18:38
@joe the rationals are a countable union of zerosets that’s not a zeroset
– Henno Brandsma
Dec 26 '18 at 18:40
In metric space every closed set is zero-set, buy you say "search about a counterexample with some closed set that is not a zero-set". I can not see.
– joe
Dec 26 '18 at 18:43
2
@joe so find a non metric $X$ as an example.
– Henno Brandsma
Dec 26 '18 at 19:00
2
@joe you can use that in a Tychonoff space all closed sets are intersections of zero-sets. Also, a zero-set is a $G_delta$ and the reverse holds in normal spaces.
– Henno Brandsma
Dec 26 '18 at 19:04
What dose it mean? singletons are in $Z(mathbb{R})$ but $ mathbb{Q}$ is not.
– joe
Dec 26 '18 at 18:38
What dose it mean? singletons are in $Z(mathbb{R})$ but $ mathbb{Q}$ is not.
– joe
Dec 26 '18 at 18:38
@joe the rationals are a countable union of zerosets that’s not a zeroset
– Henno Brandsma
Dec 26 '18 at 18:40
@joe the rationals are a countable union of zerosets that’s not a zeroset
– Henno Brandsma
Dec 26 '18 at 18:40
In metric space every closed set is zero-set, buy you say "search about a counterexample with some closed set that is not a zero-set". I can not see.
– joe
Dec 26 '18 at 18:43
In metric space every closed set is zero-set, buy you say "search about a counterexample with some closed set that is not a zero-set". I can not see.
– joe
Dec 26 '18 at 18:43
2
2
@joe so find a non metric $X$ as an example.
– Henno Brandsma
Dec 26 '18 at 19:00
@joe so find a non metric $X$ as an example.
– Henno Brandsma
Dec 26 '18 at 19:00
2
2
@joe you can use that in a Tychonoff space all closed sets are intersections of zero-sets. Also, a zero-set is a $G_delta$ and the reverse holds in normal spaces.
– Henno Brandsma
Dec 26 '18 at 19:04
@joe you can use that in a Tychonoff space all closed sets are intersections of zero-sets. Also, a zero-set is a $G_delta$ and the reverse holds in normal spaces.
– Henno Brandsma
Dec 26 '18 at 19:04
add a comment |
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2
What is $Z(X)$? What does it mean for $Z(X)$ to be, for instance, "closed under countable intersection"?
– Ben W
Dec 26 '18 at 18:15
@BenW $Z(X)$ is the set of all zero-sets of $X$.
– Henno Brandsma
Dec 26 '18 at 18:22
Hint: Find an example of a closed set that is not a zero set.
– GEdgar
Dec 26 '18 at 18:27
And what does it mean for $Z(X)$ to be closed under countable intersection?
– Ben W
Dec 26 '18 at 18:28