A question about zero-sets












2














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?











share|cite|improve this question




















  • 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














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?











share|cite|improve this question




















  • 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








2







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?











share|cite|improve this question















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






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 26 '18 at 18:24

























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














  • 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










1 Answer
1






active

oldest

votes


















0














$Z(f)$ 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.






share|cite|improve this answer





















  • 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











Your Answer





StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");

StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});

function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});


}
});














draft saved

draft discarded


















StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3053166%2fa-question-about-zero-sets%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown

























1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









0














$Z(f)$ 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.






share|cite|improve this answer





















  • 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
















0














$Z(f)$ 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.






share|cite|improve this answer





















  • 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














0












0








0






$Z(f)$ 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.






share|cite|improve this answer












$Z(f)$ 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.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










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


















  • 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


















draft saved

draft discarded




















































Thanks for contributing an answer to Mathematics Stack Exchange!


  • Please be sure to answer the question. Provide details and share your research!

But avoid



  • Asking for help, clarification, or responding to other answers.

  • Making statements based on opinion; back them up with references or personal experience.


Use MathJax to format equations. MathJax reference.


To learn more, see our tips on writing great answers.





Some of your past answers have not been well-received, and you're in danger of being blocked from answering.


Please pay close attention to the following guidance:


  • Please be sure to answer the question. Provide details and share your research!

But avoid



  • Asking for help, clarification, or responding to other answers.

  • Making statements based on opinion; back them up with references or personal experience.


To learn more, see our tips on writing great answers.




draft saved


draft discarded














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3053166%2fa-question-about-zero-sets%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown





















































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown

































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown







Popular posts from this blog

Human spaceflight

Can not write log (Is /dev/pts mounted?) - openpty in Ubuntu-on-Windows?

張江高科駅