Do continuous injections preserve open sets?












1












$begingroup$


Do continuous injections preserve open sets?



I'm pretty sure that's true in euclidean space.



If we let the singleton sets of integers generate the topology of the domain, and then identity map it to the real with standard topology, is that a counterexample?



If they don't, what combination of injective, surjective, continuous, and inverse continuous is the minimum to be an open map?



Edit: Either I was really tired and distracted on the bus when I typed this into my phone and somehow forgot to say continuous (entirely possible), or whoever put "from $mathbb{R}^m$ to $mathbb{R}^n$" in my title deleted it. Should I start a new one?






general-topology real-numbers open-map







share|cite|improve this question











$endgroup$








  • 3




    $begingroup$
    A bijective map is open iff its inverse is continuous.
    $endgroup$
    – Kavi Rama Murthy
    2 days ago










  • $begingroup$
    @KaviRamaMurthy Thanks! If we remove surjectivity, is there another condition we could add to give us an open map? And, if we add bicontinuous, can we remove anything?
    $endgroup$
    – user3146
    2 days ago
















1












$begingroup$


Do continuous injections preserve open sets?



I'm pretty sure that's true in euclidean space.



If we let the singleton sets of integers generate the topology of the domain, and then identity map it to the real with standard topology, is that a counterexample?



If they don't, what combination of injective, surjective, continuous, and inverse continuous is the minimum to be an open map?



Edit: Either I was really tired and distracted on the bus when I typed this into my phone and somehow forgot to say continuous (entirely possible), or whoever put "from $mathbb{R}^m$ to $mathbb{R}^n$" in my title deleted it. Should I start a new one?






general-topology real-numbers open-map







share|cite|improve this question











$endgroup$








  • 3




    $begingroup$
    A bijective map is open iff its inverse is continuous.
    $endgroup$
    – Kavi Rama Murthy
    2 days ago










  • $begingroup$
    @KaviRamaMurthy Thanks! If we remove surjectivity, is there another condition we could add to give us an open map? And, if we add bicontinuous, can we remove anything?
    $endgroup$
    – user3146
    2 days ago














1












1








1





$begingroup$


Do continuous injections preserve open sets?



I'm pretty sure that's true in euclidean space.



If we let the singleton sets of integers generate the topology of the domain, and then identity map it to the real with standard topology, is that a counterexample?



If they don't, what combination of injective, surjective, continuous, and inverse continuous is the minimum to be an open map?



Edit: Either I was really tired and distracted on the bus when I typed this into my phone and somehow forgot to say continuous (entirely possible), or whoever put "from $mathbb{R}^m$ to $mathbb{R}^n$" in my title deleted it. Should I start a new one?






general-topology real-numbers open-map







share|cite|improve this question











$endgroup$




Do continuous injections preserve open sets?



I'm pretty sure that's true in euclidean space.



If we let the singleton sets of integers generate the topology of the domain, and then identity map it to the real with standard topology, is that a counterexample?



If they don't, what combination of injective, surjective, continuous, and inverse continuous is the minimum to be an open map?



Edit: Either I was really tired and distracted on the bus when I typed this into my phone and somehow forgot to say continuous (entirely possible), or whoever put "from $mathbb{R}^m$ to $mathbb{R}^n$" in my title deleted it. Should I start a new one?






general-topology real-numbers open-map




general-topology real-numbers open-map






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 2 days ago







user3146

















asked 2 days ago









user3146user3146

746




746








  • 3




    $begingroup$
    A bijective map is open iff its inverse is continuous.
    $endgroup$
    – Kavi Rama Murthy
    2 days ago










  • $begingroup$
    @KaviRamaMurthy Thanks! If we remove surjectivity, is there another condition we could add to give us an open map? And, if we add bicontinuous, can we remove anything?
    $endgroup$
    – user3146
    2 days ago














  • 3




    $begingroup$
    A bijective map is open iff its inverse is continuous.
    $endgroup$
    – Kavi Rama Murthy
    2 days ago










  • $begingroup$
    @KaviRamaMurthy Thanks! If we remove surjectivity, is there another condition we could add to give us an open map? And, if we add bicontinuous, can we remove anything?
    $endgroup$
    – user3146
    2 days ago








3




3




$begingroup$
A bijective map is open iff its inverse is continuous.
$endgroup$
– Kavi Rama Murthy
2 days ago




$begingroup$
A bijective map is open iff its inverse is continuous.
$endgroup$
– Kavi Rama Murthy
2 days ago












$begingroup$
@KaviRamaMurthy Thanks! If we remove surjectivity, is there another condition we could add to give us an open map? And, if we add bicontinuous, can we remove anything?
$endgroup$
– user3146
2 days ago




$begingroup$
@KaviRamaMurthy Thanks! If we remove surjectivity, is there another condition we could add to give us an open map? And, if we add bicontinuous, can we remove anything?
$endgroup$
– user3146
2 days ago










3 Answers
3






active

oldest

votes


















11












$begingroup$

No they do not. It's not true in euclidean space.



The function $$f(x)=begin{cases}x& xleq 0\ x+1 & x>0end{cases}$$ is an injection, however, $f((-1,1)) = (-1, 0]cup (1, 2)$ which is not open.






share|cite|improve this answer









$endgroup$





















    2












    $begingroup$

    Whether $f:Xto Y$, and in particular an injection, is an open map depends on the topologies, e.g., if $Y$ has the discrete topology, then any $f$ is an open map. You can check that if $f$ is injective onto an open subset of $Y$ and $f^{-1}$ is continuous, then $f$ is an open map.






    share|cite|improve this answer









    $endgroup$





















      1












      $begingroup$

      Another example : take a set $X$ with at least two elements. Consider $X_1$ to be $X$ equiped with the discrete topology and $X_2$ to be $X$ equiped with the trivial topology. Then, the identity from $X_1$ to $X_2$ is clearly a continuous injection (it is even a bijection) but it clearly does not preserve open sets.



      I don't think there is a "minimum" requirement on a map to be open, it depends mostly on the topologies. (But homeomorphisms are always open).






      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
        });


        }
        });














        draft saved

        draft discarded


















        StackExchange.ready(
        function () {
        StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3174331%2fdo-continuous-injections-preserve-open-sets%23new-answer', 'question_page');
        }
        );

        Post as a guest















        Required, but never shown

























        3 Answers
        3






        active

        oldest

        votes








        3 Answers
        3






        active

        oldest

        votes









        active

        oldest

        votes






        active

        oldest

        votes









        11












        $begingroup$

        No they do not. It's not true in euclidean space.



        The function $$f(x)=begin{cases}x& xleq 0\ x+1 & x>0end{cases}$$ is an injection, however, $f((-1,1)) = (-1, 0]cup (1, 2)$ which is not open.






        share|cite|improve this answer









        $endgroup$


















          11












          $begingroup$

          No they do not. It's not true in euclidean space.



          The function $$f(x)=begin{cases}x& xleq 0\ x+1 & x>0end{cases}$$ is an injection, however, $f((-1,1)) = (-1, 0]cup (1, 2)$ which is not open.






          share|cite|improve this answer









          $endgroup$
















            11












            11








            11





            $begingroup$

            No they do not. It's not true in euclidean space.



            The function $$f(x)=begin{cases}x& xleq 0\ x+1 & x>0end{cases}$$ is an injection, however, $f((-1,1)) = (-1, 0]cup (1, 2)$ which is not open.






            share|cite|improve this answer









            $endgroup$



            No they do not. It's not true in euclidean space.



            The function $$f(x)=begin{cases}x& xleq 0\ x+1 & x>0end{cases}$$ is an injection, however, $f((-1,1)) = (-1, 0]cup (1, 2)$ which is not open.







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered 2 days ago









            5xum5xum

            92k394162




            92k394162























                2












                $begingroup$

                Whether $f:Xto Y$, and in particular an injection, is an open map depends on the topologies, e.g., if $Y$ has the discrete topology, then any $f$ is an open map. You can check that if $f$ is injective onto an open subset of $Y$ and $f^{-1}$ is continuous, then $f$ is an open map.






                share|cite|improve this answer









                $endgroup$


















                  2












                  $begingroup$

                  Whether $f:Xto Y$, and in particular an injection, is an open map depends on the topologies, e.g., if $Y$ has the discrete topology, then any $f$ is an open map. You can check that if $f$ is injective onto an open subset of $Y$ and $f^{-1}$ is continuous, then $f$ is an open map.






                  share|cite|improve this answer









                  $endgroup$
















                    2












                    2








                    2





                    $begingroup$

                    Whether $f:Xto Y$, and in particular an injection, is an open map depends on the topologies, e.g., if $Y$ has the discrete topology, then any $f$ is an open map. You can check that if $f$ is injective onto an open subset of $Y$ and $f^{-1}$ is continuous, then $f$ is an open map.






                    share|cite|improve this answer









                    $endgroup$



                    Whether $f:Xto Y$, and in particular an injection, is an open map depends on the topologies, e.g., if $Y$ has the discrete topology, then any $f$ is an open map. You can check that if $f$ is injective onto an open subset of $Y$ and $f^{-1}$ is continuous, then $f$ is an open map.







                    share|cite|improve this answer












                    share|cite|improve this answer



                    share|cite|improve this answer










                    answered 2 days ago









                    JosuéJosué

                    3,51242672




                    3,51242672























                        1












                        $begingroup$

                        Another example : take a set $X$ with at least two elements. Consider $X_1$ to be $X$ equiped with the discrete topology and $X_2$ to be $X$ equiped with the trivial topology. Then, the identity from $X_1$ to $X_2$ is clearly a continuous injection (it is even a bijection) but it clearly does not preserve open sets.



                        I don't think there is a "minimum" requirement on a map to be open, it depends mostly on the topologies. (But homeomorphisms are always open).






                        share|cite|improve this answer











                        $endgroup$


















                          1












                          $begingroup$

                          Another example : take a set $X$ with at least two elements. Consider $X_1$ to be $X$ equiped with the discrete topology and $X_2$ to be $X$ equiped with the trivial topology. Then, the identity from $X_1$ to $X_2$ is clearly a continuous injection (it is even a bijection) but it clearly does not preserve open sets.



                          I don't think there is a "minimum" requirement on a map to be open, it depends mostly on the topologies. (But homeomorphisms are always open).






                          share|cite|improve this answer











                          $endgroup$
















                            1












                            1








                            1





                            $begingroup$

                            Another example : take a set $X$ with at least two elements. Consider $X_1$ to be $X$ equiped with the discrete topology and $X_2$ to be $X$ equiped with the trivial topology. Then, the identity from $X_1$ to $X_2$ is clearly a continuous injection (it is even a bijection) but it clearly does not preserve open sets.



                            I don't think there is a "minimum" requirement on a map to be open, it depends mostly on the topologies. (But homeomorphisms are always open).






                            share|cite|improve this answer











                            $endgroup$



                            Another example : take a set $X$ with at least two elements. Consider $X_1$ to be $X$ equiped with the discrete topology and $X_2$ to be $X$ equiped with the trivial topology. Then, the identity from $X_1$ to $X_2$ is clearly a continuous injection (it is even a bijection) but it clearly does not preserve open sets.



                            I don't think there is a "minimum" requirement on a map to be open, it depends mostly on the topologies. (But homeomorphisms are always open).







                            share|cite|improve this answer














                            share|cite|improve this answer



                            share|cite|improve this answer








                            edited yesterday

























                            answered 2 days ago









                            L.DeRL.DeR

                            487




                            487






























                                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%2f3174331%2fdo-continuous-injections-preserve-open-sets%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

                                數位音樂下載

                                Image
                                數位音樂下載 為非以「實體」方式來販賣音樂的相關產品，而使用數位格式如mp3、AAC等文件格式來進行銷售販賣的一种音乐产业形式。著名的數位音樂下載服務商有iTunes Store、RecoChoku、Google Play音乐和微软Zune等。数字音乐下载在二十一世纪初一度风靡全球， [1] [2] 使得传统唱片产业走向衰落，但其自身也在2010年代流媒体音乐服务如Apple Music、Spotify等的挤压下迅速失去市场份额。 [3] [4] 数字音乐下载有便于携带、传播迅速、制作简单成本低等优点，与流媒体音乐相比也能更好保障音乐人权益； [5] 但它同时很可能带来侵犯版权和滥用的情形， [6] 也给音乐产业带来了诸多不利影响。数字音乐下载为音乐人和乐迷提供了更加便捷直接的交流渠道，也一定程度上推动了网络音乐人的出现和独立音乐的迅速发展。 [7] 目录 1 历史 1.1 技术积淀（1987-2000） 1.2 出现与兴起（2000-2004） 1.3 唱片市场的颠覆者（2004-2007） 1.4 巅峰和衰落（2005-2013） 1.5 摇摇欲坠的现状与不可预知的明天（2014至今） 2 主要文件格式 2.1 PCM编码格式 2.1.1 WAV 2.1.2 APE 2.1.3 FLAC 2.2 苹果推出的无损文件格式 2.2.1 AIFF 2.2.2 ALAC 2.3 有损压缩格式 2.3.1 MP3 2.3.2 AAC 3 主要市场 3.1 iTunes Store 3.2 Google Play音乐商店 3.3 亚马逊音乐商店 3.4 部分其他音乐商店 4 影响与评价 4.1 对音乐产业的影响 4.2 对科技行业的影响 4.3 正面评价 4.4 负面评价 4.4.1 侵权问题 4.4.2 经营模式不明晰 4.4.3 不适应移动时代 4.5 文化现象 5 延伸阅读 6 参考资料 历史 苹果iTunes+iPod的软硬件搭配让数字音...
                                Read more

                                When can things happen in Etherscan, such as the picture below?

                                Image
                                2 I don't know why balance of the address is 0. when can this situation? etherscan balances share | improve this question asked yesterday Ru Hi Ru Hi 11 1 New contributor Ru Hi is a new contributor to this site. Take care in asking for clarification, commenting, and answering. Check out our Code of Conduct. add a comment  |  ...
                                Read more

                                格利澤436b

                                Image
                                格利泽436b 太陽系外行星 太陽系外行星列表 艺术家笔下的格利泽436b 母恆星 母恆星 Gliese 436 星座 獅子座 赤經 ( α ) 11 h  42 m  11.0941 s [1] 赤緯 ( δ ) +26° 42′ 23.652″ [1] 距離 33.4 ly (10.2 pc) 光譜類型 M2.5 V [1] 軌道參數 半长轴 ( a ) 0.0291±0.0004 [2] AU 軌道離心率 ( e ) 0.150±0.012 [2] 公轉週期 ( P ) 2.643904±0.000005 [3] d (0.00723849 y) 軌道傾角 ( i ) 85.8 +0.21 −0.25 [3] ° 角距 ( θ ) 2.794 mas 近星點時間 ( T 0 ) 2,451,551.716 ±0.01 JD 半振幅 ( K ) 18.68±0.8 m/s 物理性质 质量 ( m ) 22.2±1.0 [2] M ⊕ 半徑 ( r ) 4.327±0.183 [2] [4] R ⊕ 密度 ( ρ ) 1.51 g cm -3 表面重力 ( g ) 1.18 g 溫度 ( T ) 712±36 [2] K 發現 發現時間 2004 發現者 巴特勒、沃格特、 馬西 et al. 發現方法 徑向速度、凌日法 發現地點 美國加利福尼亞州 發表論文 出版 其他名稱 Ross 905 b, GJ 436 b, [5] LTT 13213 b, GCTP 2704.10 b, LHS 310, AC+27:28217 b, Vyssotsky 616 b, HIP 57087 b, GEN# +9.80120068 b, LP 319-75 b, G 121-7 b, LSPM J1142+2642 b, 1RXS J114211.9+264328 b, ASCC 683818 b, G 147-68 b, UCAC2 41198281 b, BP...
                                Read more