Example Of A Non-Existent Retract












0














I am looking for an example that disproves the claim that given any subspace $A$ of a topological space $X$, there exists a retract of $X$ onto $A$.






general-topology algebraic-topology







share|cite|improve this question


















  • 3




    $S^1 subset mathbb{R}^2$
    – Tim kinsella
    Nov 26 at 1:41






  • 1




    How about the closed unit disk and the unit circle?
    – ncmathsadist
    Nov 26 at 1:41










  • Oh, duh, of course. Silly me. Thank you.
    – Frederic Chopin
    Nov 26 at 1:43










  • not silly -- its not at all obvious unless you think about it in terms of $pi_1$. $X$ retracts onto $A$ only if $pi_1(A)$ is a subgroup of $pi_1(X)$.
    – Tim kinsella
    Nov 26 at 2:07










  • That's true. But I was thinking indeed about it in terms of the fundamental group, so recalling that the homomorphism induced by the inclusion map is an injection if the space retracts onto some subspace makes this example clear immediately.
    – Frederic Chopin
    Nov 26 at 2:11


















0














I am looking for an example that disproves the claim that given any subspace $A$ of a topological space $X$, there exists a retract of $X$ onto $A$.






general-topology algebraic-topology







share|cite|improve this question


















  • 3




    $S^1 subset mathbb{R}^2$
    – Tim kinsella
    Nov 26 at 1:41






  • 1




    How about the closed unit disk and the unit circle?
    – ncmathsadist
    Nov 26 at 1:41










  • Oh, duh, of course. Silly me. Thank you.
    – Frederic Chopin
    Nov 26 at 1:43










  • not silly -- its not at all obvious unless you think about it in terms of $pi_1$. $X$ retracts onto $A$ only if $pi_1(A)$ is a subgroup of $pi_1(X)$.
    – Tim kinsella
    Nov 26 at 2:07










  • That's true. But I was thinking indeed about it in terms of the fundamental group, so recalling that the homomorphism induced by the inclusion map is an injection if the space retracts onto some subspace makes this example clear immediately.
    – Frederic Chopin
    Nov 26 at 2:11
















0












0








0


1





I am looking for an example that disproves the claim that given any subspace $A$ of a topological space $X$, there exists a retract of $X$ onto $A$.






general-topology algebraic-topology







share|cite|improve this question













I am looking for an example that disproves the claim that given any subspace $A$ of a topological space $X$, there exists a retract of $X$ onto $A$.






general-topology algebraic-topology




general-topology algebraic-topology






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Nov 26 at 1:39









Frederic Chopin

322111




322111








  • 3




    $S^1 subset mathbb{R}^2$
    – Tim kinsella
    Nov 26 at 1:41






  • 1




    How about the closed unit disk and the unit circle?
    – ncmathsadist
    Nov 26 at 1:41










  • Oh, duh, of course. Silly me. Thank you.
    – Frederic Chopin
    Nov 26 at 1:43










  • not silly -- its not at all obvious unless you think about it in terms of $pi_1$. $X$ retracts onto $A$ only if $pi_1(A)$ is a subgroup of $pi_1(X)$.
    – Tim kinsella
    Nov 26 at 2:07










  • That's true. But I was thinking indeed about it in terms of the fundamental group, so recalling that the homomorphism induced by the inclusion map is an injection if the space retracts onto some subspace makes this example clear immediately.
    – Frederic Chopin
    Nov 26 at 2:11
















  • 3




    $S^1 subset mathbb{R}^2$
    – Tim kinsella
    Nov 26 at 1:41






  • 1




    How about the closed unit disk and the unit circle?
    – ncmathsadist
    Nov 26 at 1:41










  • Oh, duh, of course. Silly me. Thank you.
    – Frederic Chopin
    Nov 26 at 1:43










  • not silly -- its not at all obvious unless you think about it in terms of $pi_1$. $X$ retracts onto $A$ only if $pi_1(A)$ is a subgroup of $pi_1(X)$.
    – Tim kinsella
    Nov 26 at 2:07










  • That's true. But I was thinking indeed about it in terms of the fundamental group, so recalling that the homomorphism induced by the inclusion map is an injection if the space retracts onto some subspace makes this example clear immediately.
    – Frederic Chopin
    Nov 26 at 2:11










3




3




$S^1 subset mathbb{R}^2$
– Tim kinsella
Nov 26 at 1:41




$S^1 subset mathbb{R}^2$
– Tim kinsella
Nov 26 at 1:41




1




1




How about the closed unit disk and the unit circle?
– ncmathsadist
Nov 26 at 1:41




How about the closed unit disk and the unit circle?
– ncmathsadist
Nov 26 at 1:41












Oh, duh, of course. Silly me. Thank you.
– Frederic Chopin
Nov 26 at 1:43




Oh, duh, of course. Silly me. Thank you.
– Frederic Chopin
Nov 26 at 1:43












not silly -- its not at all obvious unless you think about it in terms of $pi_1$. $X$ retracts onto $A$ only if $pi_1(A)$ is a subgroup of $pi_1(X)$.
– Tim kinsella
Nov 26 at 2:07




not silly -- its not at all obvious unless you think about it in terms of $pi_1$. $X$ retracts onto $A$ only if $pi_1(A)$ is a subgroup of $pi_1(X)$.
– Tim kinsella
Nov 26 at 2:07












That's true. But I was thinking indeed about it in terms of the fundamental group, so recalling that the homomorphism induced by the inclusion map is an injection if the space retracts onto some subspace makes this example clear immediately.
– Frederic Chopin
Nov 26 at 2:11






That's true. But I was thinking indeed about it in terms of the fundamental group, so recalling that the homomorphism induced by the inclusion map is an injection if the space retracts onto some subspace makes this example clear immediately.
– Frederic Chopin
Nov 26 at 2:11












2 Answers
2






active

oldest

votes


















5














Here's the simplest possible example:



Consider the space $X$ with three points $a,b,c$ and open sets $$emptyset, {a},{c}, {a,c}, {a,b,c}.$$ Let $A={a,c}$. There are only two maps from $X$ to $A$ which are the identity on $A$, and neither is continuous. E.g. if we send $b$ to $a$, then the preimage of the open set ${a}$ is the non-open set ${a,b}$.



This really is the simplest example, since any space retracts onto any of its singleton subsets and onto itself.






share|cite|improve this answer



















  • 2




    This is my favorite kind of answer to general topology questions, +1! Why get $pi_1$ involved if you don't have to? But I think $A$ should be ${a,c}$.
    – Alex Kruckman
    Nov 26 at 2:52












  • @AlexKruckman Thanks, and quite right - fixed!
    – Noah Schweber
    Nov 26 at 3:08










  • @AlexKruckman yeah, this is a better answer. but note that this is just using $pi_0$ instead of $pi_1$.
    – Tim kinsella
    Nov 29 at 6:43



















2














A retract of a Hausdorff space is closed so $(0,1)$ is not a retract of $mathbb{R}$ (usual topology).






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',
    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%2f3013691%2fexample-of-a-non-existent-retract%23new-answer', 'question_page');
    }
    );

    Post as a guest















    Required, but never shown

























    2 Answers
    2






    active

    oldest

    votes








    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    5














    Here's the simplest possible example:



    Consider the space $X$ with three points $a,b,c$ and open sets $$emptyset, {a},{c}, {a,c}, {a,b,c}.$$ Let $A={a,c}$. There are only two maps from $X$ to $A$ which are the identity on $A$, and neither is continuous. E.g. if we send $b$ to $a$, then the preimage of the open set ${a}$ is the non-open set ${a,b}$.



    This really is the simplest example, since any space retracts onto any of its singleton subsets and onto itself.






    share|cite|improve this answer



















    • 2




      This is my favorite kind of answer to general topology questions, +1! Why get $pi_1$ involved if you don't have to? But I think $A$ should be ${a,c}$.
      – Alex Kruckman
      Nov 26 at 2:52












    • @AlexKruckman Thanks, and quite right - fixed!
      – Noah Schweber
      Nov 26 at 3:08










    • @AlexKruckman yeah, this is a better answer. but note that this is just using $pi_0$ instead of $pi_1$.
      – Tim kinsella
      Nov 29 at 6:43
















    5














    Here's the simplest possible example:



    Consider the space $X$ with three points $a,b,c$ and open sets $$emptyset, {a},{c}, {a,c}, {a,b,c}.$$ Let $A={a,c}$. There are only two maps from $X$ to $A$ which are the identity on $A$, and neither is continuous. E.g. if we send $b$ to $a$, then the preimage of the open set ${a}$ is the non-open set ${a,b}$.



    This really is the simplest example, since any space retracts onto any of its singleton subsets and onto itself.






    share|cite|improve this answer



















    • 2




      This is my favorite kind of answer to general topology questions, +1! Why get $pi_1$ involved if you don't have to? But I think $A$ should be ${a,c}$.
      – Alex Kruckman
      Nov 26 at 2:52












    • @AlexKruckman Thanks, and quite right - fixed!
      – Noah Schweber
      Nov 26 at 3:08










    • @AlexKruckman yeah, this is a better answer. but note that this is just using $pi_0$ instead of $pi_1$.
      – Tim kinsella
      Nov 29 at 6:43














    5












    5








    5






    Here's the simplest possible example:



    Consider the space $X$ with three points $a,b,c$ and open sets $$emptyset, {a},{c}, {a,c}, {a,b,c}.$$ Let $A={a,c}$. There are only two maps from $X$ to $A$ which are the identity on $A$, and neither is continuous. E.g. if we send $b$ to $a$, then the preimage of the open set ${a}$ is the non-open set ${a,b}$.



    This really is the simplest example, since any space retracts onto any of its singleton subsets and onto itself.






    share|cite|improve this answer














    Here's the simplest possible example:



    Consider the space $X$ with three points $a,b,c$ and open sets $$emptyset, {a},{c}, {a,c}, {a,b,c}.$$ Let $A={a,c}$. There are only two maps from $X$ to $A$ which are the identity on $A$, and neither is continuous. E.g. if we send $b$ to $a$, then the preimage of the open set ${a}$ is the non-open set ${a,b}$.



    This really is the simplest example, since any space retracts onto any of its singleton subsets and onto itself.







    share|cite|improve this answer














    share|cite|improve this answer



    share|cite|improve this answer








    edited Nov 26 at 3:08

























    answered Nov 26 at 2:03









    Noah Schweber

    120k10146279




    120k10146279








    • 2




      This is my favorite kind of answer to general topology questions, +1! Why get $pi_1$ involved if you don't have to? But I think $A$ should be ${a,c}$.
      – Alex Kruckman
      Nov 26 at 2:52












    • @AlexKruckman Thanks, and quite right - fixed!
      – Noah Schweber
      Nov 26 at 3:08










    • @AlexKruckman yeah, this is a better answer. but note that this is just using $pi_0$ instead of $pi_1$.
      – Tim kinsella
      Nov 29 at 6:43














    • 2




      This is my favorite kind of answer to general topology questions, +1! Why get $pi_1$ involved if you don't have to? But I think $A$ should be ${a,c}$.
      – Alex Kruckman
      Nov 26 at 2:52












    • @AlexKruckman Thanks, and quite right - fixed!
      – Noah Schweber
      Nov 26 at 3:08










    • @AlexKruckman yeah, this is a better answer. but note that this is just using $pi_0$ instead of $pi_1$.
      – Tim kinsella
      Nov 29 at 6:43








    2




    2




    This is my favorite kind of answer to general topology questions, +1! Why get $pi_1$ involved if you don't have to? But I think $A$ should be ${a,c}$.
    – Alex Kruckman
    Nov 26 at 2:52






    This is my favorite kind of answer to general topology questions, +1! Why get $pi_1$ involved if you don't have to? But I think $A$ should be ${a,c}$.
    – Alex Kruckman
    Nov 26 at 2:52














    @AlexKruckman Thanks, and quite right - fixed!
    – Noah Schweber
    Nov 26 at 3:08




    @AlexKruckman Thanks, and quite right - fixed!
    – Noah Schweber
    Nov 26 at 3:08












    @AlexKruckman yeah, this is a better answer. but note that this is just using $pi_0$ instead of $pi_1$.
    – Tim kinsella
    Nov 29 at 6:43




    @AlexKruckman yeah, this is a better answer. but note that this is just using $pi_0$ instead of $pi_1$.
    – Tim kinsella
    Nov 29 at 6:43











    2














    A retract of a Hausdorff space is closed so $(0,1)$ is not a retract of $mathbb{R}$ (usual topology).






    share|cite|improve this answer


























      2














      A retract of a Hausdorff space is closed so $(0,1)$ is not a retract of $mathbb{R}$ (usual topology).






      share|cite|improve this answer
























        2












        2








        2






        A retract of a Hausdorff space is closed so $(0,1)$ is not a retract of $mathbb{R}$ (usual topology).






        share|cite|improve this answer












        A retract of a Hausdorff space is closed so $(0,1)$ is not a retract of $mathbb{R}$ (usual topology).







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Nov 26 at 6:17









        Henno Brandsma

        104k346113




        104k346113






























            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%2f3013691%2fexample-of-a-non-existent-retract%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

            Bundesstraße 106

            Image
            Dieser Artikel beschreibt die Bundesstraße 106 in Deutschland. Zur gleichnamigen Straße in Österreich siehe Mölltal Straße. Vorlage:Infobox hochrangige Straße/Wartung/DE-B Bundesstraße 106 in Deutschland Karte Basisdaten Betreiber: Deutschland  Bundesrepublik Deutschland Straßenbeginn: Wismar ( 53° 54′  N , 11° 24′  O 53.900536 11.396875 ) Straßenende: Schwerin ( 53° 36′  N , 11° 24′  O 53.592785 11.404892 ) Gesamtlänge: 40,7 km Bundesland: Mecklenburg-Vorpommern Ausbauzustand: zweistreifig [1] Bundesstraße 106 in Schwerin (Umgehungsstraße) Straßenverlauf Land Mecklenburg-Vorpommern Landkreis Nordwestmecklenburg Ortsumgehung Wismar Wallensteingraben über   Dorf Mecklenburg Wallensteingraben Groß Stieten Bad Kleinen OT Niendorf Bahnstrecke Lübeck–Bad Kleinen Zickhusen Lübstorf Klein Trebbow OT Kirch Stück
            Read more

            Verónica Boquete

            Image
            Verónica Boquete im Trikot der Nationalmannschaft (2013) Personalia Name Verónica Boquete Giadans Geburtstag 9. April 1987 Geburtsort Santiago de Compostela, Spanien Größe 159 cm Position Mittelfeld / Sturm Juniorinnen Jahre Station 2001–2005 Xuventú Aguiño Frauen Jahre Station Spiele (Tore) 1 2005–2008 Prainsa Zaragoza 83 (40) 2008–2010 Espanyol Barcelona 53 (30) 2010 Buffalo Flash 9 0 (9) 2010 Chicago Red Stars 3 0 (1) 2010–2011 Espanyol Barcelona 26 (41) 2011 Philadelphia Independence 13 0 (5) 2011 Energija Woronesch 5 0 (1) 2012–2014 Tyresö FF 39 (14) 2014 Portland Thorns FC 15 0 (4) 2014–2015 1. FFC Frankfurt 23 0 (7) 2015–2016 FC Bayern München 9 0 (0) 2016–2018 Paris Saint-Germain 31 0 (8) 2018 Peking BG Phoenix 2019– Utah Royals FC 0 0 (0) Nationalmannschaft Jahre Auswahl
            Read more

            Ida-Boy-Ed-Garten

            Image
            Die Lage des Ida-Boy-Ed-Gartens, rot markiert Der Ida-Boy-Ed-Garten Der Ida-Boy-Ed-Garten, Blick von der Wakenitzmauer durch das Tor in der Stadtmauer Der Ida-Boy-Ed-Garten ist eine Straße der Lübecker Altstadt. Inhaltsverzeichnis 1 Lage 2 Geschichte 3 Literatur 4 Weblinks Lage | Der etwa 90 Meter lange Ida-Boy-Ed-Garten befindet sich an der nördlichen Spitze der Altstadtinsel, vor dem Burgtor. Die in einem Bogen durch eine Grünanlage hangaufwärts führende Straße verbindet die Wakenitzmauer mit der Burgtorbrücke. Geschichte | Der heutige Ida-Boy-Ed-Garten wurde ursprünglich als Verlängerung der Wakenitzmauer angelegt. 1903 wurde in den erhaltenen Teil der mittelalterlichen Stadtmauer, an dem die Wakenitzmauer bei der Einmündung der Kaiserstraße begann, ein Tor mit einem großen Bogen für die Durchführung der Straße und zwei kleineren seitlichen Öffnungen für Fußgänger gebrochen, um für die neu errichteten Wohnhäuser in diesem Bereich
            Read more