Bicontinuous bijection between line and squareIs there a continuous bijection between an interval $[0,1]$ and a square: $[0,1] times [0,1]$?Bijection between $[0,1]$ and $K$Bijection between measurable setsBijection between open and closed intervalPath From Positive Dedekind Cuts to Reals?Bijection between closed unit interval and $ R$Strong Counterexample to MVT on QStrong Counterexample to MVT on QHow do we know that there are not more numbers than there are names?Direct bijection between $C[0,1]$ and $[0,1]$bijection between the Cantor set and $R$

Multi tool use
Multi tool use

How to get directions in deep space?

Can I cause damage to electrical appliances by unplugging them when they are turned on?

Anime with legendary swords made from talismans and a man who could change them with a shattered body

Isometric embedding of a genus g surface

How to leave product feedback on macOS?

Do people actually use the word "kaputt" in conversation?

Quoting Keynes in a lecture

What should be the ideal length of sentences in a blog post for ease of reading?

Did I make a mistake by ccing email to boss to others?

Why is the principal energy of an electron lower for excited electrons in a higher energy state?

How do I Interface a PS/2 Keyboard without Modern Techniques?

Why is participating in the European Parliamentary elections used as a threat?

Personal or impersonal in a technical resume

Is there a reason to prefer HFS+ over APFS for disk images in High Sierra and/or Mojave?

How to reduce predictors the right way for a logistic regression model

Why does the Persian emissary display a string of crowned skulls?

Sound waves in different octaves

Giving feedback to someone without sounding prejudiced

How can I safely use "Thalidomide" in my novel while respecting the trademark?

What (the heck) is a Super Worm Equinox Moon?

Limit max CPU usage SQL SERVER with WSRM

"Oh no!" in Latin

What does "Scientists rise up against statistical significance" mean? (Comment in Nature)

Are Captain Marvel's powers affected by Thanos breaking the Tesseract and claiming the stone?



Bicontinuous bijection between line and square


Is there a continuous bijection between an interval $[0,1]$ and a square: $[0,1] times [0,1]$?Bijection between $[0,1]$ and $K$Bijection between measurable setsBijection between open and closed intervalPath From Positive Dedekind Cuts to Reals?Bijection between closed unit interval and $ R$Strong Counterexample to MVT on QStrong Counterexample to MVT on QHow do we know that there are not more numbers than there are names?Direct bijection between $C[0,1]$ and $[0,1]$bijection between the Cantor set and $R$













0












$begingroup$


A few years ago I saw a reference to two articles by two prominent 19th Century mathematicians, whose names I can no longer remember, to constructions of bicontinuous bijections between an interval, maybe open, maybe closed, of the reals, and a square, similarly open or closed, in 2-dim. Euclidean space, which I wouldn't have believed possible. However, I couldn't find any access to the articles. Can anyone provide me with a description of such a construction, or a way to access such an article?



On the other hand, can anyone provide me with a proof, or a way to access a proof, that such a construction is impossible?










share|cite|improve this question







New contributor




Michael Fox is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$











  • $begingroup$
    Do you mean this?
    $endgroup$
    – poyea
    Mar 14 at 8:07















0












$begingroup$


A few years ago I saw a reference to two articles by two prominent 19th Century mathematicians, whose names I can no longer remember, to constructions of bicontinuous bijections between an interval, maybe open, maybe closed, of the reals, and a square, similarly open or closed, in 2-dim. Euclidean space, which I wouldn't have believed possible. However, I couldn't find any access to the articles. Can anyone provide me with a description of such a construction, or a way to access such an article?



On the other hand, can anyone provide me with a proof, or a way to access a proof, that such a construction is impossible?










share|cite|improve this question







New contributor




Michael Fox is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$











  • $begingroup$
    Do you mean this?
    $endgroup$
    – poyea
    Mar 14 at 8:07













0












0








0





$begingroup$


A few years ago I saw a reference to two articles by two prominent 19th Century mathematicians, whose names I can no longer remember, to constructions of bicontinuous bijections between an interval, maybe open, maybe closed, of the reals, and a square, similarly open or closed, in 2-dim. Euclidean space, which I wouldn't have believed possible. However, I couldn't find any access to the articles. Can anyone provide me with a description of such a construction, or a way to access such an article?



On the other hand, can anyone provide me with a proof, or a way to access a proof, that such a construction is impossible?










share|cite|improve this question







New contributor




Michael Fox is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$




A few years ago I saw a reference to two articles by two prominent 19th Century mathematicians, whose names I can no longer remember, to constructions of bicontinuous bijections between an interval, maybe open, maybe closed, of the reals, and a square, similarly open or closed, in 2-dim. Euclidean space, which I wouldn't have believed possible. However, I couldn't find any access to the articles. Can anyone provide me with a description of such a construction, or a way to access such an article?



On the other hand, can anyone provide me with a proof, or a way to access a proof, that such a construction is impossible?







real-analysis






share|cite|improve this question







New contributor




Michael Fox is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











share|cite|improve this question







New contributor




Michael Fox is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









share|cite|improve this question




share|cite|improve this question






New contributor




Michael Fox is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









asked Mar 14 at 8:04









Michael FoxMichael Fox

13




13




New contributor




Michael Fox is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.





New contributor





Michael Fox is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






Michael Fox is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











  • $begingroup$
    Do you mean this?
    $endgroup$
    – poyea
    Mar 14 at 8:07
















  • $begingroup$
    Do you mean this?
    $endgroup$
    – poyea
    Mar 14 at 8:07















$begingroup$
Do you mean this?
$endgroup$
– poyea
Mar 14 at 8:07




$begingroup$
Do you mean this?
$endgroup$
– poyea
Mar 14 at 8:07










1 Answer
1






active

oldest

votes


















0












$begingroup$

No such thing is possible. if you remove a point in the middle of the interval it becomes disconnected. If you remove a point from a square it remains connected. So no such function can exist.






share|cite|improve this answer









$endgroup$












    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
    );



    );






    Michael Fox is a new contributor. Be nice, and check out our Code of Conduct.









    draft saved

    draft discarded


















    StackExchange.ready(
    function ()
    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3147701%2fbicontinuous-bijection-between-line-and-square%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












    $begingroup$

    No such thing is possible. if you remove a point in the middle of the interval it becomes disconnected. If you remove a point from a square it remains connected. So no such function can exist.






    share|cite|improve this answer









    $endgroup$

















      0












      $begingroup$

      No such thing is possible. if you remove a point in the middle of the interval it becomes disconnected. If you remove a point from a square it remains connected. So no such function can exist.






      share|cite|improve this answer









      $endgroup$















        0












        0








        0





        $begingroup$

        No such thing is possible. if you remove a point in the middle of the interval it becomes disconnected. If you remove a point from a square it remains connected. So no such function can exist.






        share|cite|improve this answer









        $endgroup$



        No such thing is possible. if you remove a point in the middle of the interval it becomes disconnected. If you remove a point from a square it remains connected. So no such function can exist.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Mar 14 at 8:07









        Kavi Rama MurthyKavi Rama Murthy

        68.7k53169




        68.7k53169




















            Michael Fox is a new contributor. Be nice, and check out our Code of Conduct.









            draft saved

            draft discarded


















            Michael Fox is a new contributor. Be nice, and check out our Code of Conduct.












            Michael Fox is a new contributor. Be nice, and check out our Code of Conduct.











            Michael Fox is a new contributor. Be nice, and check out our Code of Conduct.














            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%2f3147701%2fbicontinuous-bijection-between-line-and-square%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







            cDnXdcCRKO9yL 9xfr7OfwmR P,ke7yXjnJ,7gs aW56xoteX SFii5Z0 w5w 3HinGjwPBOfNouN 7,E4Esfz
            AXa0R sg0I DNYXt C77YXE0ycs ASYV OFkt8a3 00I5p5wCfmwy C5,Sspd,L4pkcoK pH,WgAYrx3BCEcLIoKy4Yj41Pp2J

            Popular posts from this blog

            Football at the 1986 Brunei Merdeka Games Contents Teams Group stage Knockout stage References Navigation menu"Brunei Merdeka Games 1986".

            Solar Wings Breeze Design and development Specifications (Breeze) References Navigation menu1368-485X"Hang glider: Breeze (Solar Wings)"e

            Kathakali Contents Etymology and nomenclature History Repertoire Songs and musical instruments Traditional plays Styles: Sampradayam Training centers and awards Relationship to other dance forms See also Notes References External links Navigation menueThe Illustrated Encyclopedia of Hinduism: A-MSouth Asian Folklore: An EncyclopediaRoutledge International Encyclopedia of Women: Global Women's Issues and KnowledgeKathakali Dance-drama: Where Gods and Demons Come to PlayKathakali Dance-drama: Where Gods and Demons Come to PlayKathakali Dance-drama: Where Gods and Demons Come to Play10.1353/atj.2005.0004The Illustrated Encyclopedia of Hinduism: A-MEncyclopedia of HinduismKathakali Dance-drama: Where Gods and Demons Come to PlaySonic Liturgy: Ritual and Music in Hindu Tradition"The Mirror of Gesture"Kathakali Dance-drama: Where Gods and Demons Come to Play"Kathakali"Indian Theatre: Traditions of PerformanceIndian Theatre: Traditions of PerformanceIndian Theatre: Traditions of PerformanceIndian Theatre: Traditions of PerformanceMedieval Indian Literature: An AnthologyThe Oxford Companion to Indian TheatreSouth Asian Folklore: An Encyclopedia : Afghanistan, Bangladesh, India, Nepal, Pakistan, Sri LankaThe Rise of Performance Studies: Rethinking Richard Schechner's Broad SpectrumIndian Theatre: Traditions of PerformanceModern Asian Theatre and Performance 1900-2000Critical Theory and PerformanceBetween Theater and AnthropologyKathakali603847011Indian Theatre: Traditions of PerformanceIndian Theatre: Traditions of PerformanceIndian Theatre: Traditions of PerformanceBetween Theater and AnthropologyBetween Theater and AnthropologyNambeesan Smaraka AwardsArchivedThe Cambridge Guide to TheatreRoutledge International Encyclopedia of Women: Global Women's Issues and KnowledgeThe Garland Encyclopedia of World Music: South Asia : the Indian subcontinentThe Ethos of Noh: Actors and Their Art10.2307/1145740By Means of Performance: Intercultural Studies of Theatre and Ritual10.1017/s204912550000100xReconceiving the Renaissance: A Critical ReaderPerformance TheoryListening to Theatre: The Aural Dimension of Beijing Opera10.2307/1146013Kathakali: The Art of the Non-WorldlyOn KathakaliKathakali, the dance theatreThe Kathakali Complex: Performance & StructureKathakali Dance-Drama: Where Gods and Demons Come to Play10.1093/obo/9780195399318-0071Drama and Ritual of Early Hinduism"In the Shadow of Hollywood Orientalism: Authentic East Indian Dancing"10.1080/08949460490274013Sanskrit Play Production in Ancient IndiaIndian Music: History and StructureBharata, the Nāṭyaśāstra233639306Table of Contents2238067286469807Dance In Indian Painting10.2307/32047833204783Kathakali Dance-Theatre: A Visual Narrative of Sacred Indian MimeIndian Classical Dance: The Renaissance and BeyondKathakali: an indigenous art-form of Keralaeee