Is $0$ a cluster point of the set ${0}$?












3












$begingroup$


I just read the theorem




A subset of R is closed if and only if it contains all of its cluster points.




And I found that the set {0} is closed (Is that correct?), then the set {0} should contains all its cluster points, right? But I just checked the definition of cluster point in the textbook (it appears at the beginning of the limit of function) and found that 0 is not the cluster point of the set {0}



I want to know where the false is ?






real-analysis real-numbers







share|cite|improve this question











$endgroup$












  • $begingroup$
    Do you think that the set is not closed? If yes then can you give a cluster point which is not in the set? The definition says that A set is closed iff it contains all its cluster point. If 0 is not the cluster point then why do you even worry about it?
    $endgroup$
    – Rakesh Bhatt
    Dec 15 '18 at 6:51










  • $begingroup$
    I got it ! I just thought 0 "should" be a cluster point of the set .Then the set does not have a cluster point, right? Thank you. But wait a minute, the set {0} contains the element 0 !
    $endgroup$
    – Yang Gao
    Dec 15 '18 at 7:06








  • 3




    $begingroup$
    Post the definition of "cluster point" in your book.
    $endgroup$
    – xbh
    Dec 15 '18 at 7:12










  • $begingroup$
    I think it appears at the answer provided by Anthony Ter .
    $endgroup$
    – Yang Gao
    Dec 15 '18 at 7:19










  • $begingroup$
    @YangGao The set {0} contains the element 0! Yes, It contains. So what?
    $endgroup$
    – Rakesh Bhatt
    Dec 15 '18 at 9:21
















3












$begingroup$


I just read the theorem




A subset of R is closed if and only if it contains all of its cluster points.




And I found that the set {0} is closed (Is that correct?), then the set {0} should contains all its cluster points, right? But I just checked the definition of cluster point in the textbook (it appears at the beginning of the limit of function) and found that 0 is not the cluster point of the set {0}



I want to know where the false is ?






real-analysis real-numbers







share|cite|improve this question











$endgroup$












  • $begingroup$
    Do you think that the set is not closed? If yes then can you give a cluster point which is not in the set? The definition says that A set is closed iff it contains all its cluster point. If 0 is not the cluster point then why do you even worry about it?
    $endgroup$
    – Rakesh Bhatt
    Dec 15 '18 at 6:51










  • $begingroup$
    I got it ! I just thought 0 "should" be a cluster point of the set .Then the set does not have a cluster point, right? Thank you. But wait a minute, the set {0} contains the element 0 !
    $endgroup$
    – Yang Gao
    Dec 15 '18 at 7:06








  • 3




    $begingroup$
    Post the definition of "cluster point" in your book.
    $endgroup$
    – xbh
    Dec 15 '18 at 7:12










  • $begingroup$
    I think it appears at the answer provided by Anthony Ter .
    $endgroup$
    – Yang Gao
    Dec 15 '18 at 7:19










  • $begingroup$
    @YangGao The set {0} contains the element 0! Yes, It contains. So what?
    $endgroup$
    – Rakesh Bhatt
    Dec 15 '18 at 9:21














3












3








3


2



$begingroup$


I just read the theorem




A subset of R is closed if and only if it contains all of its cluster points.




And I found that the set {0} is closed (Is that correct?), then the set {0} should contains all its cluster points, right? But I just checked the definition of cluster point in the textbook (it appears at the beginning of the limit of function) and found that 0 is not the cluster point of the set {0}



I want to know where the false is ?






real-analysis real-numbers







share|cite|improve this question











$endgroup$




I just read the theorem




A subset of R is closed if and only if it contains all of its cluster points.




And I found that the set {0} is closed (Is that correct?), then the set {0} should contains all its cluster points, right? But I just checked the definition of cluster point in the textbook (it appears at the beginning of the limit of function) and found that 0 is not the cluster point of the set {0}



I want to know where the false is ?






real-analysis real-numbers




real-analysis real-numbers






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 15 '18 at 7:13









anomaly

17.8k42666




17.8k42666










asked Dec 15 '18 at 6:48









Yang GaoYang Gao

186




186












  • $begingroup$
    Do you think that the set is not closed? If yes then can you give a cluster point which is not in the set? The definition says that A set is closed iff it contains all its cluster point. If 0 is not the cluster point then why do you even worry about it?
    $endgroup$
    – Rakesh Bhatt
    Dec 15 '18 at 6:51










  • $begingroup$
    I got it ! I just thought 0 "should" be a cluster point of the set .Then the set does not have a cluster point, right? Thank you. But wait a minute, the set {0} contains the element 0 !
    $endgroup$
    – Yang Gao
    Dec 15 '18 at 7:06








  • 3




    $begingroup$
    Post the definition of "cluster point" in your book.
    $endgroup$
    – xbh
    Dec 15 '18 at 7:12










  • $begingroup$
    I think it appears at the answer provided by Anthony Ter .
    $endgroup$
    – Yang Gao
    Dec 15 '18 at 7:19










  • $begingroup$
    @YangGao The set {0} contains the element 0! Yes, It contains. So what?
    $endgroup$
    – Rakesh Bhatt
    Dec 15 '18 at 9:21


















  • $begingroup$
    Do you think that the set is not closed? If yes then can you give a cluster point which is not in the set? The definition says that A set is closed iff it contains all its cluster point. If 0 is not the cluster point then why do you even worry about it?
    $endgroup$
    – Rakesh Bhatt
    Dec 15 '18 at 6:51










  • $begingroup$
    I got it ! I just thought 0 "should" be a cluster point of the set .Then the set does not have a cluster point, right? Thank you. But wait a minute, the set {0} contains the element 0 !
    $endgroup$
    – Yang Gao
    Dec 15 '18 at 7:06








  • 3




    $begingroup$
    Post the definition of "cluster point" in your book.
    $endgroup$
    – xbh
    Dec 15 '18 at 7:12










  • $begingroup$
    I think it appears at the answer provided by Anthony Ter .
    $endgroup$
    – Yang Gao
    Dec 15 '18 at 7:19










  • $begingroup$
    @YangGao The set {0} contains the element 0! Yes, It contains. So what?
    $endgroup$
    – Rakesh Bhatt
    Dec 15 '18 at 9:21
















$begingroup$
Do you think that the set is not closed? If yes then can you give a cluster point which is not in the set? The definition says that A set is closed iff it contains all its cluster point. If 0 is not the cluster point then why do you even worry about it?
$endgroup$
– Rakesh Bhatt
Dec 15 '18 at 6:51




$begingroup$
Do you think that the set is not closed? If yes then can you give a cluster point which is not in the set? The definition says that A set is closed iff it contains all its cluster point. If 0 is not the cluster point then why do you even worry about it?
$endgroup$
– Rakesh Bhatt
Dec 15 '18 at 6:51












$begingroup$
I got it ! I just thought 0 "should" be a cluster point of the set .Then the set does not have a cluster point, right? Thank you. But wait a minute, the set {0} contains the element 0 !
$endgroup$
– Yang Gao
Dec 15 '18 at 7:06






$begingroup$
I got it ! I just thought 0 "should" be a cluster point of the set .Then the set does not have a cluster point, right? Thank you. But wait a minute, the set {0} contains the element 0 !
$endgroup$
– Yang Gao
Dec 15 '18 at 7:06






3




3




$begingroup$
Post the definition of "cluster point" in your book.
$endgroup$
– xbh
Dec 15 '18 at 7:12




$begingroup$
Post the definition of "cluster point" in your book.
$endgroup$
– xbh
Dec 15 '18 at 7:12












$begingroup$
I think it appears at the answer provided by Anthony Ter .
$endgroup$
– Yang Gao
Dec 15 '18 at 7:19




$begingroup$
I think it appears at the answer provided by Anthony Ter .
$endgroup$
– Yang Gao
Dec 15 '18 at 7:19












$begingroup$
@YangGao The set {0} contains the element 0! Yes, It contains. So what?
$endgroup$
– Rakesh Bhatt
Dec 15 '18 at 9:21




$begingroup$
@YangGao The set {0} contains the element 0! Yes, It contains. So what?
$endgroup$
– Rakesh Bhatt
Dec 15 '18 at 9:21










1 Answer
1






active

oldest

votes


















2












$begingroup$

If $0$ was a cluster point of ${0}$ then for every $delta > 0$ there would exist an $x neq 0, x in N_delta(0)$ such that $x in {0}$. clearly no such $x$ exists, for any $delta$. Thus $0$ is not a cluster point of ${0}$.



So is ${0 }$ closed? Does it contain all of its limit points? Well, $emptyset subseteq {0}$, so ${0}$ is closed.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Thanks. Now I really get it . The closed set contains all of its cluster point but not all of its elements! Thank you!
    $endgroup$
    – Yang Gao
    Dec 15 '18 at 7:17










  • $begingroup$
    Now the empty set is the only cluster point of the set, right? Just a little "weird".
    $endgroup$
    – Yang Gao
    Dec 15 '18 at 7:25






  • 1




    $begingroup$
    No,the empty set isn't a point. The empty set represents the set of limit points, and I was trying to show that that set is contained in ${0}$. I should have used $subseteq$, not $in$. I'll edit it now.
    $endgroup$
    – Anthony Ter
    Dec 15 '18 at 7:27








  • 1




    $begingroup$
    So the set does not have a cluster point ? wow
    $endgroup$
    – Yang Gao
    Dec 15 '18 at 7:31










  • $begingroup$
    Yep, in fact, no finite subset of $mathbb{R}$ has a cluster point.
    $endgroup$
    – Anthony Ter
    Dec 15 '18 at 7:35












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%2f3040240%2fis-0-a-cluster-point-of-the-set-0%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









2












$begingroup$

If $0$ was a cluster point of ${0}$ then for every $delta > 0$ there would exist an $x neq 0, x in N_delta(0)$ such that $x in {0}$. clearly no such $x$ exists, for any $delta$. Thus $0$ is not a cluster point of ${0}$.



So is ${0 }$ closed? Does it contain all of its limit points? Well, $emptyset subseteq {0}$, so ${0}$ is closed.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Thanks. Now I really get it . The closed set contains all of its cluster point but not all of its elements! Thank you!
    $endgroup$
    – Yang Gao
    Dec 15 '18 at 7:17










  • $begingroup$
    Now the empty set is the only cluster point of the set, right? Just a little "weird".
    $endgroup$
    – Yang Gao
    Dec 15 '18 at 7:25






  • 1




    $begingroup$
    No,the empty set isn't a point. The empty set represents the set of limit points, and I was trying to show that that set is contained in ${0}$. I should have used $subseteq$, not $in$. I'll edit it now.
    $endgroup$
    – Anthony Ter
    Dec 15 '18 at 7:27








  • 1




    $begingroup$
    So the set does not have a cluster point ? wow
    $endgroup$
    – Yang Gao
    Dec 15 '18 at 7:31










  • $begingroup$
    Yep, in fact, no finite subset of $mathbb{R}$ has a cluster point.
    $endgroup$
    – Anthony Ter
    Dec 15 '18 at 7:35
















2












$begingroup$

If $0$ was a cluster point of ${0}$ then for every $delta > 0$ there would exist an $x neq 0, x in N_delta(0)$ such that $x in {0}$. clearly no such $x$ exists, for any $delta$. Thus $0$ is not a cluster point of ${0}$.



So is ${0 }$ closed? Does it contain all of its limit points? Well, $emptyset subseteq {0}$, so ${0}$ is closed.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Thanks. Now I really get it . The closed set contains all of its cluster point but not all of its elements! Thank you!
    $endgroup$
    – Yang Gao
    Dec 15 '18 at 7:17










  • $begingroup$
    Now the empty set is the only cluster point of the set, right? Just a little "weird".
    $endgroup$
    – Yang Gao
    Dec 15 '18 at 7:25






  • 1




    $begingroup$
    No,the empty set isn't a point. The empty set represents the set of limit points, and I was trying to show that that set is contained in ${0}$. I should have used $subseteq$, not $in$. I'll edit it now.
    $endgroup$
    – Anthony Ter
    Dec 15 '18 at 7:27








  • 1




    $begingroup$
    So the set does not have a cluster point ? wow
    $endgroup$
    – Yang Gao
    Dec 15 '18 at 7:31










  • $begingroup$
    Yep, in fact, no finite subset of $mathbb{R}$ has a cluster point.
    $endgroup$
    – Anthony Ter
    Dec 15 '18 at 7:35














2












2








2





$begingroup$

If $0$ was a cluster point of ${0}$ then for every $delta > 0$ there would exist an $x neq 0, x in N_delta(0)$ such that $x in {0}$. clearly no such $x$ exists, for any $delta$. Thus $0$ is not a cluster point of ${0}$.



So is ${0 }$ closed? Does it contain all of its limit points? Well, $emptyset subseteq {0}$, so ${0}$ is closed.






share|cite|improve this answer











$endgroup$



If $0$ was a cluster point of ${0}$ then for every $delta > 0$ there would exist an $x neq 0, x in N_delta(0)$ such that $x in {0}$. clearly no such $x$ exists, for any $delta$. Thus $0$ is not a cluster point of ${0}$.



So is ${0 }$ closed? Does it contain all of its limit points? Well, $emptyset subseteq {0}$, so ${0}$ is closed.







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Dec 15 '18 at 7:28

























answered Dec 15 '18 at 7:04









Anthony TerAnthony Ter

39616




39616












  • $begingroup$
    Thanks. Now I really get it . The closed set contains all of its cluster point but not all of its elements! Thank you!
    $endgroup$
    – Yang Gao
    Dec 15 '18 at 7:17










  • $begingroup$
    Now the empty set is the only cluster point of the set, right? Just a little "weird".
    $endgroup$
    – Yang Gao
    Dec 15 '18 at 7:25






  • 1




    $begingroup$
    No,the empty set isn't a point. The empty set represents the set of limit points, and I was trying to show that that set is contained in ${0}$. I should have used $subseteq$, not $in$. I'll edit it now.
    $endgroup$
    – Anthony Ter
    Dec 15 '18 at 7:27








  • 1




    $begingroup$
    So the set does not have a cluster point ? wow
    $endgroup$
    – Yang Gao
    Dec 15 '18 at 7:31










  • $begingroup$
    Yep, in fact, no finite subset of $mathbb{R}$ has a cluster point.
    $endgroup$
    – Anthony Ter
    Dec 15 '18 at 7:35


















  • $begingroup$
    Thanks. Now I really get it . The closed set contains all of its cluster point but not all of its elements! Thank you!
    $endgroup$
    – Yang Gao
    Dec 15 '18 at 7:17










  • $begingroup$
    Now the empty set is the only cluster point of the set, right? Just a little "weird".
    $endgroup$
    – Yang Gao
    Dec 15 '18 at 7:25






  • 1




    $begingroup$
    No,the empty set isn't a point. The empty set represents the set of limit points, and I was trying to show that that set is contained in ${0}$. I should have used $subseteq$, not $in$. I'll edit it now.
    $endgroup$
    – Anthony Ter
    Dec 15 '18 at 7:27








  • 1




    $begingroup$
    So the set does not have a cluster point ? wow
    $endgroup$
    – Yang Gao
    Dec 15 '18 at 7:31










  • $begingroup$
    Yep, in fact, no finite subset of $mathbb{R}$ has a cluster point.
    $endgroup$
    – Anthony Ter
    Dec 15 '18 at 7:35
















$begingroup$
Thanks. Now I really get it . The closed set contains all of its cluster point but not all of its elements! Thank you!
$endgroup$
– Yang Gao
Dec 15 '18 at 7:17




$begingroup$
Thanks. Now I really get it . The closed set contains all of its cluster point but not all of its elements! Thank you!
$endgroup$
– Yang Gao
Dec 15 '18 at 7:17












$begingroup$
Now the empty set is the only cluster point of the set, right? Just a little "weird".
$endgroup$
– Yang Gao
Dec 15 '18 at 7:25




$begingroup$
Now the empty set is the only cluster point of the set, right? Just a little "weird".
$endgroup$
– Yang Gao
Dec 15 '18 at 7:25




1




1




$begingroup$
No,the empty set isn't a point. The empty set represents the set of limit points, and I was trying to show that that set is contained in ${0}$. I should have used $subseteq$, not $in$. I'll edit it now.
$endgroup$
– Anthony Ter
Dec 15 '18 at 7:27






$begingroup$
No,the empty set isn't a point. The empty set represents the set of limit points, and I was trying to show that that set is contained in ${0}$. I should have used $subseteq$, not $in$. I'll edit it now.
$endgroup$
– Anthony Ter
Dec 15 '18 at 7:27






1




1




$begingroup$
So the set does not have a cluster point ? wow
$endgroup$
– Yang Gao
Dec 15 '18 at 7:31




$begingroup$
So the set does not have a cluster point ? wow
$endgroup$
– Yang Gao
Dec 15 '18 at 7:31












$begingroup$
Yep, in fact, no finite subset of $mathbb{R}$ has a cluster point.
$endgroup$
– Anthony Ter
Dec 15 '18 at 7:35




$begingroup$
Yep, in fact, no finite subset of $mathbb{R}$ has a cluster point.
$endgroup$
– Anthony Ter
Dec 15 '18 at 7:35


















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.




draft saved


draft discarded














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3040240%2fis-0-a-cluster-point-of-the-set-0%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

How to change which sound is reproduced for terminal bell?

Image
3 Recently upgrade to 18.10 from 18.04, and using the terminal is becoming annoying: the new alert sound is a drip ( /usr/share/sounds/gnome/default/alerts/drip.ogg ), and the previous one is still located at ( /usr/share/sounds/ubuntu/stereo/bell.ogg ). My issue is that at the gnome-terminal, when autocompleting or pressing left BOTH sounds are reproduced, at the same time. I want to just have ubuntu/stereo/bell.ogg active, not both . At the moment I have been only able to reduce the volume of the alerts (but still both are played), or turn the alerts down (but I dislike both "solutions"). gnome-terminal 18.10 teminal-bell share | improve this question
Read more

Title Spacing in Bjornstrup Chapter, Removing Chapter Number From Contents

Image
up vote 5 down vote favorite 2 I would like to make it so that the vertical space below the title of each chapter is the same when using the Bjornstrup package (i.e. so that the descenders do not affect the spacing - as shown by the red arrow in the picture below). I think I'm fairly happy with the spacing around "A chapter", so I'd really like titles without descenders (e.g. "Contents") to have the same vertical spacing as "A chapter" below the title. I would also like to remove the large "0" in the contents title. Any help appreciated, as always. documentclass[11pt,a4paper]{book} % Packages[![enter image description here][1]][1] usepackage[left=3.5cm, right=3.5cm, top=4.3cm, bottom=4.25cm]{geometry} usepackage[utf8]{inputenc} usepackage{libertine} usepackage[libertine]{newtx
Read more

Can I use Tabulator js library in my java Spring + Thymeleaf project?

Image
0 I want to add Tabulator js to my java project. I added tabulator.min.js, tabulator.min.css to resources folder. After that added to my html page these dependencies as recommending in readme.md. <link rel="stylesheet" th:href="@{/css/tabulator.min.css}"> <script type="text/javascript" th:src="@{/js/tabulator.min.js}"></script> Now my tables view all elements, that I have in the html page, but not correct. In the documentation of Tabulator is example how to create header and table, var table = new Tabulator("#example-table", { columns:[ {title:"Name", field:"name", sorter:"string", width:200, editor:true}, {title:"Age", field:"age", sorter:"
Read more