$bigcup U_{i}$ and $bigcup V_{j}$ are homeomorphic iff there are equally many $U_{i}$ and $V_{j}$ where...












1












$begingroup$


Let $U$ and $V$ two open sets in $mathbb{R}$.




  • First, I proved that if $U$ is open, then $U = bigsqcup_{i in mathbb{N}} U_{i}$ where $U_{i}$ is open and $"bigsqcup"$ means disjoint union.


  • Second, I proved that $U_{i}$ are uniquely determined by $U$, that is, $bigsqcup_{i in mathbb{N}} U_{i}$ is unique, up to order of the factors.



Now, I want to prove that




If $U$ and $V$ are open, $U$ and $V$ are homeomorphic iff there are equally many $U_{i}$ and $V_{j}$.




This makes perfect sense for me, but I dont know how to write correctly.



Can someone help me?



I dont know if it helps, but I wrote $U = bigsqcup I_{x}$ where $I_{x}$ is the largest interval containing $x$ and countained in $U$.






real-analysis general-topology metric-spaces







share|cite|improve this question









$endgroup$

















    1












    $begingroup$


    Let $U$ and $V$ two open sets in $mathbb{R}$.




    • First, I proved that if $U$ is open, then $U = bigsqcup_{i in mathbb{N}} U_{i}$ where $U_{i}$ is open and $"bigsqcup"$ means disjoint union.


    • Second, I proved that $U_{i}$ are uniquely determined by $U$, that is, $bigsqcup_{i in mathbb{N}} U_{i}$ is unique, up to order of the factors.



    Now, I want to prove that




    If $U$ and $V$ are open, $U$ and $V$ are homeomorphic iff there are equally many $U_{i}$ and $V_{j}$.




    This makes perfect sense for me, but I dont know how to write correctly.



    Can someone help me?



    I dont know if it helps, but I wrote $U = bigsqcup I_{x}$ where $I_{x}$ is the largest interval containing $x$ and countained in $U$.






    real-analysis general-topology metric-spaces







    share|cite|improve this question









    $endgroup$















      1












      1








      1





      $begingroup$


      Let $U$ and $V$ two open sets in $mathbb{R}$.




      • First, I proved that if $U$ is open, then $U = bigsqcup_{i in mathbb{N}} U_{i}$ where $U_{i}$ is open and $"bigsqcup"$ means disjoint union.


      • Second, I proved that $U_{i}$ are uniquely determined by $U$, that is, $bigsqcup_{i in mathbb{N}} U_{i}$ is unique, up to order of the factors.



      Now, I want to prove that




      If $U$ and $V$ are open, $U$ and $V$ are homeomorphic iff there are equally many $U_{i}$ and $V_{j}$.




      This makes perfect sense for me, but I dont know how to write correctly.



      Can someone help me?



      I dont know if it helps, but I wrote $U = bigsqcup I_{x}$ where $I_{x}$ is the largest interval containing $x$ and countained in $U$.






      real-analysis general-topology metric-spaces







      share|cite|improve this question









      $endgroup$




      Let $U$ and $V$ two open sets in $mathbb{R}$.




      • First, I proved that if $U$ is open, then $U = bigsqcup_{i in mathbb{N}} U_{i}$ where $U_{i}$ is open and $"bigsqcup"$ means disjoint union.


      • Second, I proved that $U_{i}$ are uniquely determined by $U$, that is, $bigsqcup_{i in mathbb{N}} U_{i}$ is unique, up to order of the factors.



      Now, I want to prove that




      If $U$ and $V$ are open, $U$ and $V$ are homeomorphic iff there are equally many $U_{i}$ and $V_{j}$.




      This makes perfect sense for me, but I dont know how to write correctly.



      Can someone help me?



      I dont know if it helps, but I wrote $U = bigsqcup I_{x}$ where $I_{x}$ is the largest interval containing $x$ and countained in $U$.






      real-analysis general-topology metric-spaces




      real-analysis general-topology metric-spaces






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Dec 20 '18 at 20:45









      Lucas CorrêaLucas Corrêa

      1,6251321




      1,6251321






















          1 Answer
          1






          active

          oldest

          votes


















          1












          $begingroup$

          Hints: your construction $U = bigcup_x I_x$, where $I_x$ is a maximal interval containing $x in U$, is a good way of describing the decomposition of $U$ into its connected components, but you need to pick just one $x$ from each connected component to get a disjoint union. You are right that there will be at most countably many distinct connected components $I_x$, but there need not be infinitely many (e.g., take $U = Bbb{R}$). So your first observation should be that $U = bigsqcup_{j in J}U_j$, where $J subseteq Bbb{N}$ and each $U_j$ is non-empty open interval. Now if you have $V = bigsqcup_{k in K}V_k$ where $K subseteq Bbb{N}$ and each $V_k$ is a non-empty open interval, then, if $J$ and $K$ have the same number of elements you can use a bijection between $J$ and $K$ and your favourite homeomomorphism between any two open intervals $(a, b)$ and $(c, d)$ ($a < b$ and $c < d)$ to construct a homeomorphism between $U$ and $V$. Conversely you can use a homeomorphism between $U$ and $V$ to construct a bijection between $J$ and $K$ (so $J$ and $K$ have "equally many elements").






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            I got the first hint. For converse, I try to take a homeomorphism between $U$ and $V$ and take the restrictions $f|_{U_{i}}: U_{i} to A$. Since $f$ is open, $A$ must be an open set. So, how can I show that $J$ and $K$ are "equal"?
            $endgroup$
            – Lucas Corrêa
            Dec 21 '18 at 1:43










          • $begingroup$
            The homeomorphism must give you a bijection between the connected components of $U$ and $V$ and thus a bijection between the sets $J$ and $K$ that we are using to index the connected components.
            $endgroup$
            – Rob Arthan
            Dec 21 '18 at 1:56











          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%2f3047952%2fbigcup-u-i-and-bigcup-v-j-are-homeomorphic-iff-there-are-equally-many%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









          1












          $begingroup$

          Hints: your construction $U = bigcup_x I_x$, where $I_x$ is a maximal interval containing $x in U$, is a good way of describing the decomposition of $U$ into its connected components, but you need to pick just one $x$ from each connected component to get a disjoint union. You are right that there will be at most countably many distinct connected components $I_x$, but there need not be infinitely many (e.g., take $U = Bbb{R}$). So your first observation should be that $U = bigsqcup_{j in J}U_j$, where $J subseteq Bbb{N}$ and each $U_j$ is non-empty open interval. Now if you have $V = bigsqcup_{k in K}V_k$ where $K subseteq Bbb{N}$ and each $V_k$ is a non-empty open interval, then, if $J$ and $K$ have the same number of elements you can use a bijection between $J$ and $K$ and your favourite homeomomorphism between any two open intervals $(a, b)$ and $(c, d)$ ($a < b$ and $c < d)$ to construct a homeomorphism between $U$ and $V$. Conversely you can use a homeomorphism between $U$ and $V$ to construct a bijection between $J$ and $K$ (so $J$ and $K$ have "equally many elements").






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            I got the first hint. For converse, I try to take a homeomorphism between $U$ and $V$ and take the restrictions $f|_{U_{i}}: U_{i} to A$. Since $f$ is open, $A$ must be an open set. So, how can I show that $J$ and $K$ are "equal"?
            $endgroup$
            – Lucas Corrêa
            Dec 21 '18 at 1:43










          • $begingroup$
            The homeomorphism must give you a bijection between the connected components of $U$ and $V$ and thus a bijection between the sets $J$ and $K$ that we are using to index the connected components.
            $endgroup$
            – Rob Arthan
            Dec 21 '18 at 1:56
















          1












          $begingroup$

          Hints: your construction $U = bigcup_x I_x$, where $I_x$ is a maximal interval containing $x in U$, is a good way of describing the decomposition of $U$ into its connected components, but you need to pick just one $x$ from each connected component to get a disjoint union. You are right that there will be at most countably many distinct connected components $I_x$, but there need not be infinitely many (e.g., take $U = Bbb{R}$). So your first observation should be that $U = bigsqcup_{j in J}U_j$, where $J subseteq Bbb{N}$ and each $U_j$ is non-empty open interval. Now if you have $V = bigsqcup_{k in K}V_k$ where $K subseteq Bbb{N}$ and each $V_k$ is a non-empty open interval, then, if $J$ and $K$ have the same number of elements you can use a bijection between $J$ and $K$ and your favourite homeomomorphism between any two open intervals $(a, b)$ and $(c, d)$ ($a < b$ and $c < d)$ to construct a homeomorphism between $U$ and $V$. Conversely you can use a homeomorphism between $U$ and $V$ to construct a bijection between $J$ and $K$ (so $J$ and $K$ have "equally many elements").






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            I got the first hint. For converse, I try to take a homeomorphism between $U$ and $V$ and take the restrictions $f|_{U_{i}}: U_{i} to A$. Since $f$ is open, $A$ must be an open set. So, how can I show that $J$ and $K$ are "equal"?
            $endgroup$
            – Lucas Corrêa
            Dec 21 '18 at 1:43










          • $begingroup$
            The homeomorphism must give you a bijection between the connected components of $U$ and $V$ and thus a bijection between the sets $J$ and $K$ that we are using to index the connected components.
            $endgroup$
            – Rob Arthan
            Dec 21 '18 at 1:56














          1












          1








          1





          $begingroup$

          Hints: your construction $U = bigcup_x I_x$, where $I_x$ is a maximal interval containing $x in U$, is a good way of describing the decomposition of $U$ into its connected components, but you need to pick just one $x$ from each connected component to get a disjoint union. You are right that there will be at most countably many distinct connected components $I_x$, but there need not be infinitely many (e.g., take $U = Bbb{R}$). So your first observation should be that $U = bigsqcup_{j in J}U_j$, where $J subseteq Bbb{N}$ and each $U_j$ is non-empty open interval. Now if you have $V = bigsqcup_{k in K}V_k$ where $K subseteq Bbb{N}$ and each $V_k$ is a non-empty open interval, then, if $J$ and $K$ have the same number of elements you can use a bijection between $J$ and $K$ and your favourite homeomomorphism between any two open intervals $(a, b)$ and $(c, d)$ ($a < b$ and $c < d)$ to construct a homeomorphism between $U$ and $V$. Conversely you can use a homeomorphism between $U$ and $V$ to construct a bijection between $J$ and $K$ (so $J$ and $K$ have "equally many elements").






          share|cite|improve this answer









          $endgroup$



          Hints: your construction $U = bigcup_x I_x$, where $I_x$ is a maximal interval containing $x in U$, is a good way of describing the decomposition of $U$ into its connected components, but you need to pick just one $x$ from each connected component to get a disjoint union. You are right that there will be at most countably many distinct connected components $I_x$, but there need not be infinitely many (e.g., take $U = Bbb{R}$). So your first observation should be that $U = bigsqcup_{j in J}U_j$, where $J subseteq Bbb{N}$ and each $U_j$ is non-empty open interval. Now if you have $V = bigsqcup_{k in K}V_k$ where $K subseteq Bbb{N}$ and each $V_k$ is a non-empty open interval, then, if $J$ and $K$ have the same number of elements you can use a bijection between $J$ and $K$ and your favourite homeomomorphism between any two open intervals $(a, b)$ and $(c, d)$ ($a < b$ and $c < d)$ to construct a homeomorphism between $U$ and $V$. Conversely you can use a homeomorphism between $U$ and $V$ to construct a bijection between $J$ and $K$ (so $J$ and $K$ have "equally many elements").







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 20 '18 at 21:27









          Rob ArthanRob Arthan

          29.2k42966




          29.2k42966












          • $begingroup$
            I got the first hint. For converse, I try to take a homeomorphism between $U$ and $V$ and take the restrictions $f|_{U_{i}}: U_{i} to A$. Since $f$ is open, $A$ must be an open set. So, how can I show that $J$ and $K$ are "equal"?
            $endgroup$
            – Lucas Corrêa
            Dec 21 '18 at 1:43










          • $begingroup$
            The homeomorphism must give you a bijection between the connected components of $U$ and $V$ and thus a bijection between the sets $J$ and $K$ that we are using to index the connected components.
            $endgroup$
            – Rob Arthan
            Dec 21 '18 at 1:56


















          • $begingroup$
            I got the first hint. For converse, I try to take a homeomorphism between $U$ and $V$ and take the restrictions $f|_{U_{i}}: U_{i} to A$. Since $f$ is open, $A$ must be an open set. So, how can I show that $J$ and $K$ are "equal"?
            $endgroup$
            – Lucas Corrêa
            Dec 21 '18 at 1:43










          • $begingroup$
            The homeomorphism must give you a bijection between the connected components of $U$ and $V$ and thus a bijection between the sets $J$ and $K$ that we are using to index the connected components.
            $endgroup$
            – Rob Arthan
            Dec 21 '18 at 1:56
















          $begingroup$
          I got the first hint. For converse, I try to take a homeomorphism between $U$ and $V$ and take the restrictions $f|_{U_{i}}: U_{i} to A$. Since $f$ is open, $A$ must be an open set. So, how can I show that $J$ and $K$ are "equal"?
          $endgroup$
          – Lucas Corrêa
          Dec 21 '18 at 1:43




          $begingroup$
          I got the first hint. For converse, I try to take a homeomorphism between $U$ and $V$ and take the restrictions $f|_{U_{i}}: U_{i} to A$. Since $f$ is open, $A$ must be an open set. So, how can I show that $J$ and $K$ are "equal"?
          $endgroup$
          – Lucas Corrêa
          Dec 21 '18 at 1:43












          $begingroup$
          The homeomorphism must give you a bijection between the connected components of $U$ and $V$ and thus a bijection between the sets $J$ and $K$ that we are using to index the connected components.
          $endgroup$
          – Rob Arthan
          Dec 21 '18 at 1:56




          $begingroup$
          The homeomorphism must give you a bijection between the connected components of $U$ and $V$ and thus a bijection between the sets $J$ and $K$ that we are using to index the connected components.
          $endgroup$
          – Rob Arthan
          Dec 21 '18 at 1:56


















          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%2f3047952%2fbigcup-u-i-and-bigcup-v-j-are-homeomorphic-iff-there-are-equally-many%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

          Bressuire

          Image
          .mw-parser-output .infobox{border:1px solid #aaa;background-color:#f9f9f9;color:black;margin:.5em 0 .5em 1em;padding:.2em;float:right;clear:right;width:22em;text-align:left;font-size:88%;line-height:1.6em}.mw-parser-output .infobox td,.mw-parser-output .infobox th{vertical-align:top;padding:0 .2em}.mw-parser-output .infobox caption{font-size:larger}.mw-parser-output .infobox.bordered{border-collapse:collapse}.mw-parser-output .infobox.bordered td,.mw-parser-output .infobox.bordered th{border:1px solid #aaa}.mw-parser-output .infobox.bordered .borderless td,.mw-parser-output .infobox.bordered .borderless th{border:0}.mw-parser-output .infobox-showbutton .mw-collapsible-text{color:inherit}.mw-parser-output .infobox.bordered .mergedtoprow td,.mw-parser-output .infobox.bordered .mergedtoprow th{border:0;border-top:1px solid #aaa;border-right:1px solid #aaa}.mw-parser-output .infobox.bordered .mergedrow td,.mw-parser-output .infobox.bordered .mergedrow th{border:0;border-right:1px solid ...
          Read more

          Cabo Verde

          Image
            Nota: Cabo-verdiano redireciona para este artigo. Para a língua falada em Cabo Verde, veja crioulo cabo-verdiano. Para outros significados, veja Cabo Verde (desambiguação). Coordenadas: 14° 55' N 23° 31' O Cabo Verde República de Cabo Verde Bandeira Brasão de Armas Lema: Unidade, Trabalho, Progresso Hino nacional: Cântico da Liberdade Gentílico: cabo-verdiano [ 1 ] Capital Praia 14° 55' N 23° 31' O Cidade mais populosa Praia Língua oficial Português Governo República unitária semipresidencialista [ 2 ]  - Presidente Jorge Carlos Fonseca  - Primeiro-ministro Ulisses Correia e Silva História    - Descoberta 1460   - Independência 00 • de Portugal 5 de Julho de 1975   - Multipartidarismo 13 de Janeiro de 1990  Área    - Total 4 033 km² (146.º)  - Água (%) 0,08 População    - Estimativa para 2017 560 899 [ 3 ] hab. (173.º) ...
          Read more

          Gyllenstierna

          Image
          .mw-parser-output .infobox{border:1px solid #aaa;background-color:#f9f9f9;color:black;margin:.5em 0 .5em 1em;padding:.2em;float:right;clear:right;width:22em;text-align:left;font-size:88%;line-height:1.6em}.mw-parser-output .infobox td,.mw-parser-output .infobox th{vertical-align:top;padding:0 .2em}.mw-parser-output .infobox caption{font-size:larger}.mw-parser-output .infobox.bordered{border-collapse:collapse}.mw-parser-output .infobox.bordered td,.mw-parser-output .infobox.bordered th{border:1px solid #aaa}.mw-parser-output .infobox.bordered .borderless td,.mw-parser-output .infobox.bordered .borderless th{border:0}.mw-parser-output .infobox-showbutton .mw-collapsible-text{color:inherit}.mw-parser-output .infobox.bordered .mergedtoprow td,.mw-parser-output .infobox.bordered .mergedtoprow th{border:0;border-top:1px solid #aaa;border-right:1px solid #aaa}.mw-parser-output .infobox.bordered .mergedrow td,.mw-parser-output .infobox.bordered .mergedrow th{border:0;border-right:1px solid ...
          Read more