What is the difference between “closed ” and “bounded” in terms of domains?











up vote
1
down vote

favorite












I'm working on understanding double & triple integrals (and 3-space geometry in general) and I often encounter in my textbooks the requirement that f is continuous on "a closed bounded domain". Wouldn't these concepts be the same thing? Like, if a domain is closed, it contains it's endpoints, and it thus necessarily finite, and if it is bounded it is contained within some "ball" of finite radius centered around the origin and is so finite. I can't really imagine a domain being closed, and not bound, or vice versa. Am I missing some detail in distinguishing these two?










share|cite|improve this question


















  • 1




    $mathbb{R}$ is closed, but not bounded
    – qwenty
    Mar 15 '15 at 11:01










  • Furthermore, $(0,1)$ is bounded, but not closed.
    – Relure
    Mar 15 '15 at 11:04










  • In $mathbf R^n$, a closed bounded domain is exactly a compact domain, it it has many wonderful properties: a numerical function has a maximum and a minimum, every sequence has a convergent subsequence, &c.
    – Bernard
    Mar 15 '15 at 11:10










  • The plane ${y = x}$, the sphere ${|mathbf x| = 1}$, the graph of any unbounded function $f(x,y)$ defined over $Bbb R^2$. All of these are closed but not bounded! (as subsets of $Bbb R^3$)
    – GPerez
    Mar 15 '15 at 12:24












  • @user2715119 So it is "closed", in the sense of being closed under addition and multiplication (as well as satisfying the other axioms of vector spaces)? Not closed as in "interval containing its endpoints"? Since $mathbb{R}$ is not a closed interval, I assume you are referring to the first part? So a closed geometric shape is one where the set of all points contained "inside" the shape, is closed?
    – Laplacinator
    Mar 15 '15 at 12:40

















up vote
1
down vote

favorite












I'm working on understanding double & triple integrals (and 3-space geometry in general) and I often encounter in my textbooks the requirement that f is continuous on "a closed bounded domain". Wouldn't these concepts be the same thing? Like, if a domain is closed, it contains it's endpoints, and it thus necessarily finite, and if it is bounded it is contained within some "ball" of finite radius centered around the origin and is so finite. I can't really imagine a domain being closed, and not bound, or vice versa. Am I missing some detail in distinguishing these two?










share|cite|improve this question


















  • 1




    $mathbb{R}$ is closed, but not bounded
    – qwenty
    Mar 15 '15 at 11:01










  • Furthermore, $(0,1)$ is bounded, but not closed.
    – Relure
    Mar 15 '15 at 11:04










  • In $mathbf R^n$, a closed bounded domain is exactly a compact domain, it it has many wonderful properties: a numerical function has a maximum and a minimum, every sequence has a convergent subsequence, &c.
    – Bernard
    Mar 15 '15 at 11:10










  • The plane ${y = x}$, the sphere ${|mathbf x| = 1}$, the graph of any unbounded function $f(x,y)$ defined over $Bbb R^2$. All of these are closed but not bounded! (as subsets of $Bbb R^3$)
    – GPerez
    Mar 15 '15 at 12:24












  • @user2715119 So it is "closed", in the sense of being closed under addition and multiplication (as well as satisfying the other axioms of vector spaces)? Not closed as in "interval containing its endpoints"? Since $mathbb{R}$ is not a closed interval, I assume you are referring to the first part? So a closed geometric shape is one where the set of all points contained "inside" the shape, is closed?
    – Laplacinator
    Mar 15 '15 at 12:40















up vote
1
down vote

favorite









up vote
1
down vote

favorite











I'm working on understanding double & triple integrals (and 3-space geometry in general) and I often encounter in my textbooks the requirement that f is continuous on "a closed bounded domain". Wouldn't these concepts be the same thing? Like, if a domain is closed, it contains it's endpoints, and it thus necessarily finite, and if it is bounded it is contained within some "ball" of finite radius centered around the origin and is so finite. I can't really imagine a domain being closed, and not bound, or vice versa. Am I missing some detail in distinguishing these two?










share|cite|improve this question













I'm working on understanding double & triple integrals (and 3-space geometry in general) and I often encounter in my textbooks the requirement that f is continuous on "a closed bounded domain". Wouldn't these concepts be the same thing? Like, if a domain is closed, it contains it's endpoints, and it thus necessarily finite, and if it is bounded it is contained within some "ball" of finite radius centered around the origin and is so finite. I can't really imagine a domain being closed, and not bound, or vice versa. Am I missing some detail in distinguishing these two?







calculus definite-integrals






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Mar 15 '15 at 10:58









Laplacinator

470513




470513








  • 1




    $mathbb{R}$ is closed, but not bounded
    – qwenty
    Mar 15 '15 at 11:01










  • Furthermore, $(0,1)$ is bounded, but not closed.
    – Relure
    Mar 15 '15 at 11:04










  • In $mathbf R^n$, a closed bounded domain is exactly a compact domain, it it has many wonderful properties: a numerical function has a maximum and a minimum, every sequence has a convergent subsequence, &c.
    – Bernard
    Mar 15 '15 at 11:10










  • The plane ${y = x}$, the sphere ${|mathbf x| = 1}$, the graph of any unbounded function $f(x,y)$ defined over $Bbb R^2$. All of these are closed but not bounded! (as subsets of $Bbb R^3$)
    – GPerez
    Mar 15 '15 at 12:24












  • @user2715119 So it is "closed", in the sense of being closed under addition and multiplication (as well as satisfying the other axioms of vector spaces)? Not closed as in "interval containing its endpoints"? Since $mathbb{R}$ is not a closed interval, I assume you are referring to the first part? So a closed geometric shape is one where the set of all points contained "inside" the shape, is closed?
    – Laplacinator
    Mar 15 '15 at 12:40
















  • 1




    $mathbb{R}$ is closed, but not bounded
    – qwenty
    Mar 15 '15 at 11:01










  • Furthermore, $(0,1)$ is bounded, but not closed.
    – Relure
    Mar 15 '15 at 11:04










  • In $mathbf R^n$, a closed bounded domain is exactly a compact domain, it it has many wonderful properties: a numerical function has a maximum and a minimum, every sequence has a convergent subsequence, &c.
    – Bernard
    Mar 15 '15 at 11:10










  • The plane ${y = x}$, the sphere ${|mathbf x| = 1}$, the graph of any unbounded function $f(x,y)$ defined over $Bbb R^2$. All of these are closed but not bounded! (as subsets of $Bbb R^3$)
    – GPerez
    Mar 15 '15 at 12:24












  • @user2715119 So it is "closed", in the sense of being closed under addition and multiplication (as well as satisfying the other axioms of vector spaces)? Not closed as in "interval containing its endpoints"? Since $mathbb{R}$ is not a closed interval, I assume you are referring to the first part? So a closed geometric shape is one where the set of all points contained "inside" the shape, is closed?
    – Laplacinator
    Mar 15 '15 at 12:40










1




1




$mathbb{R}$ is closed, but not bounded
– qwenty
Mar 15 '15 at 11:01




$mathbb{R}$ is closed, but not bounded
– qwenty
Mar 15 '15 at 11:01












Furthermore, $(0,1)$ is bounded, but not closed.
– Relure
Mar 15 '15 at 11:04




Furthermore, $(0,1)$ is bounded, but not closed.
– Relure
Mar 15 '15 at 11:04












In $mathbf R^n$, a closed bounded domain is exactly a compact domain, it it has many wonderful properties: a numerical function has a maximum and a minimum, every sequence has a convergent subsequence, &c.
– Bernard
Mar 15 '15 at 11:10




In $mathbf R^n$, a closed bounded domain is exactly a compact domain, it it has many wonderful properties: a numerical function has a maximum and a minimum, every sequence has a convergent subsequence, &c.
– Bernard
Mar 15 '15 at 11:10












The plane ${y = x}$, the sphere ${|mathbf x| = 1}$, the graph of any unbounded function $f(x,y)$ defined over $Bbb R^2$. All of these are closed but not bounded! (as subsets of $Bbb R^3$)
– GPerez
Mar 15 '15 at 12:24






The plane ${y = x}$, the sphere ${|mathbf x| = 1}$, the graph of any unbounded function $f(x,y)$ defined over $Bbb R^2$. All of these are closed but not bounded! (as subsets of $Bbb R^3$)
– GPerez
Mar 15 '15 at 12:24














@user2715119 So it is "closed", in the sense of being closed under addition and multiplication (as well as satisfying the other axioms of vector spaces)? Not closed as in "interval containing its endpoints"? Since $mathbb{R}$ is not a closed interval, I assume you are referring to the first part? So a closed geometric shape is one where the set of all points contained "inside" the shape, is closed?
– Laplacinator
Mar 15 '15 at 12:40






@user2715119 So it is "closed", in the sense of being closed under addition and multiplication (as well as satisfying the other axioms of vector spaces)? Not closed as in "interval containing its endpoints"? Since $mathbb{R}$ is not a closed interval, I assume you are referring to the first part? So a closed geometric shape is one where the set of all points contained "inside" the shape, is closed?
– Laplacinator
Mar 15 '15 at 12:40












1 Answer
1






active

oldest

votes

















up vote
0
down vote













In a bounded set, the endpoints need not necessarily be a part of the set whereas in a closed set, the endpoints need to be a part of that set (as you have mentioned in your question). E.g. [0,1] and [0,1) are both bounded (by 0 and 1), but the second set isn't closed.






share|cite|improve this answer





















    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',
    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%2f1190640%2fwhat-is-the-difference-between-closed-and-bounded-in-terms-of-domains%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








    up vote
    0
    down vote













    In a bounded set, the endpoints need not necessarily be a part of the set whereas in a closed set, the endpoints need to be a part of that set (as you have mentioned in your question). E.g. [0,1] and [0,1) are both bounded (by 0 and 1), but the second set isn't closed.






    share|cite|improve this answer

























      up vote
      0
      down vote













      In a bounded set, the endpoints need not necessarily be a part of the set whereas in a closed set, the endpoints need to be a part of that set (as you have mentioned in your question). E.g. [0,1] and [0,1) are both bounded (by 0 and 1), but the second set isn't closed.






      share|cite|improve this answer























        up vote
        0
        down vote










        up vote
        0
        down vote









        In a bounded set, the endpoints need not necessarily be a part of the set whereas in a closed set, the endpoints need to be a part of that set (as you have mentioned in your question). E.g. [0,1] and [0,1) are both bounded (by 0 and 1), but the second set isn't closed.






        share|cite|improve this answer












        In a bounded set, the endpoints need not necessarily be a part of the set whereas in a closed set, the endpoints need to be a part of that set (as you have mentioned in your question). E.g. [0,1] and [0,1) are both bounded (by 0 and 1), but the second set isn't closed.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Oct 22 at 13:39









        Arpan Ganguli

        113




        113






























            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%2f1190640%2fwhat-is-the-difference-between-closed-and-bounded-in-terms-of-domains%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

            Ellipse (mathématiques)

            Quarter-circle Tiles

            Mont Emei