Is a connected set always an uncountably infinite set?












6












$begingroup$


I'm trying to understand the concept of a connected set. The classic example which is presented is that of $Bbb R$ or any interval of $Bbb R$ with the usual topology.

Moreover, I heard that to a certain extent connected sets can be considered opposite to discrete sets. They are sometimes indicated as representing the idea of a continuum. So, intuitively, shouldn't they always be uncountable sets?






real-analysis general-topology connectedness







share|cite|improve this question











$endgroup$








  • 6




    $begingroup$
    It's plainly false that every subset of $mathbb{R}$ is connected. The connected subsets of $mathbb{R}$ are the empty set, the singletons and all intervals. But $[0,1]cup[2,3]$ is obviously disconnected.
    $endgroup$
    – egreg
    Feb 3 at 19:14












  • $begingroup$
    @egreg You're right. I meant any interval :)
    $endgroup$
    – Gabriele Scarlatti
    Feb 3 at 23:21






  • 1




    $begingroup$
    @GabrieleScarlatti You might want to edit your question accordingly.
    $endgroup$
    – Luke
    Feb 5 at 19:08










  • $begingroup$
    @Luke I"ve edited it :)
    $endgroup$
    – Gabriele Scarlatti
    Feb 6 at 12:03


















6












$begingroup$


I'm trying to understand the concept of a connected set. The classic example which is presented is that of $Bbb R$ or any interval of $Bbb R$ with the usual topology.

Moreover, I heard that to a certain extent connected sets can be considered opposite to discrete sets. They are sometimes indicated as representing the idea of a continuum. So, intuitively, shouldn't they always be uncountable sets?






real-analysis general-topology connectedness







share|cite|improve this question











$endgroup$








  • 6




    $begingroup$
    It's plainly false that every subset of $mathbb{R}$ is connected. The connected subsets of $mathbb{R}$ are the empty set, the singletons and all intervals. But $[0,1]cup[2,3]$ is obviously disconnected.
    $endgroup$
    – egreg
    Feb 3 at 19:14












  • $begingroup$
    @egreg You're right. I meant any interval :)
    $endgroup$
    – Gabriele Scarlatti
    Feb 3 at 23:21






  • 1




    $begingroup$
    @GabrieleScarlatti You might want to edit your question accordingly.
    $endgroup$
    – Luke
    Feb 5 at 19:08










  • $begingroup$
    @Luke I"ve edited it :)
    $endgroup$
    – Gabriele Scarlatti
    Feb 6 at 12:03
















6












6








6





$begingroup$


I'm trying to understand the concept of a connected set. The classic example which is presented is that of $Bbb R$ or any interval of $Bbb R$ with the usual topology.

Moreover, I heard that to a certain extent connected sets can be considered opposite to discrete sets. They are sometimes indicated as representing the idea of a continuum. So, intuitively, shouldn't they always be uncountable sets?






real-analysis general-topology connectedness







share|cite|improve this question











$endgroup$




I'm trying to understand the concept of a connected set. The classic example which is presented is that of $Bbb R$ or any interval of $Bbb R$ with the usual topology.

Moreover, I heard that to a certain extent connected sets can be considered opposite to discrete sets. They are sometimes indicated as representing the idea of a continuum. So, intuitively, shouldn't they always be uncountable sets?






real-analysis general-topology connectedness




real-analysis general-topology connectedness






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Feb 6 at 11:59







Gabriele Scarlatti

















asked Feb 3 at 17:58









Gabriele ScarlattiGabriele Scarlatti

380212




380212








  • 6




    $begingroup$
    It's plainly false that every subset of $mathbb{R}$ is connected. The connected subsets of $mathbb{R}$ are the empty set, the singletons and all intervals. But $[0,1]cup[2,3]$ is obviously disconnected.
    $endgroup$
    – egreg
    Feb 3 at 19:14












  • $begingroup$
    @egreg You're right. I meant any interval :)
    $endgroup$
    – Gabriele Scarlatti
    Feb 3 at 23:21






  • 1




    $begingroup$
    @GabrieleScarlatti You might want to edit your question accordingly.
    $endgroup$
    – Luke
    Feb 5 at 19:08










  • $begingroup$
    @Luke I"ve edited it :)
    $endgroup$
    – Gabriele Scarlatti
    Feb 6 at 12:03
















  • 6




    $begingroup$
    It's plainly false that every subset of $mathbb{R}$ is connected. The connected subsets of $mathbb{R}$ are the empty set, the singletons and all intervals. But $[0,1]cup[2,3]$ is obviously disconnected.
    $endgroup$
    – egreg
    Feb 3 at 19:14












  • $begingroup$
    @egreg You're right. I meant any interval :)
    $endgroup$
    – Gabriele Scarlatti
    Feb 3 at 23:21






  • 1




    $begingroup$
    @GabrieleScarlatti You might want to edit your question accordingly.
    $endgroup$
    – Luke
    Feb 5 at 19:08










  • $begingroup$
    @Luke I"ve edited it :)
    $endgroup$
    – Gabriele Scarlatti
    Feb 6 at 12:03










6




6




$begingroup$
It's plainly false that every subset of $mathbb{R}$ is connected. The connected subsets of $mathbb{R}$ are the empty set, the singletons and all intervals. But $[0,1]cup[2,3]$ is obviously disconnected.
$endgroup$
– egreg
Feb 3 at 19:14






$begingroup$
It's plainly false that every subset of $mathbb{R}$ is connected. The connected subsets of $mathbb{R}$ are the empty set, the singletons and all intervals. But $[0,1]cup[2,3]$ is obviously disconnected.
$endgroup$
– egreg
Feb 3 at 19:14














$begingroup$
@egreg You're right. I meant any interval :)
$endgroup$
– Gabriele Scarlatti
Feb 3 at 23:21




$begingroup$
@egreg You're right. I meant any interval :)
$endgroup$
– Gabriele Scarlatti
Feb 3 at 23:21




1




1




$begingroup$
@GabrieleScarlatti You might want to edit your question accordingly.
$endgroup$
– Luke
Feb 5 at 19:08




$begingroup$
@GabrieleScarlatti You might want to edit your question accordingly.
$endgroup$
– Luke
Feb 5 at 19:08












$begingroup$
@Luke I"ve edited it :)
$endgroup$
– Gabriele Scarlatti
Feb 6 at 12:03






$begingroup$
@Luke I"ve edited it :)
$endgroup$
– Gabriele Scarlatti
Feb 6 at 12:03












3 Answers
3






active

oldest

votes


















18












$begingroup$

First of all, $emptyset$ is connected. And so is every singleton.



On the other hand, it is true that every connected metric space with more than one point is uncountable. But there are countable connected topological spaces.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    I've heard some argue that the empty set is not connected, in a similar sense to 1 not being a prime number. See ncatlab.org/nlab/show/too+simple+to+be+simple for related thoughts.
    $endgroup$
    – Ben Millwood
    Feb 4 at 12:54










  • $begingroup$
    @BenMillwood, to me, the empty set is probably connected, but not $0$-connected (because $0$-connected implies $(-1)$-connected, and the latter is equivalent to being non-empty).
    $endgroup$
    – Carsten S
    Feb 4 at 14:14



















10












$begingroup$

Consider the space $X = {a, b}$ equipped with the trivial topology (i.e. the only open sets are $X$ and $varnothing$). Then, the space is connected, but there are only a finite number of points in this example.






share|cite|improve this answer









$endgroup$









  • 9




    $begingroup$
    Even cooler is the pseudocircle. It's the smallest multiconnected space.
    $endgroup$
    – PyRulez
    Feb 3 at 23:10



















5












$begingroup$

Well, there's the empty set, as others have noted. Also, every singleton is connected. But you can say a bit more if you assume some separation.




  • Any finite $T_1$ space is discrete, so it is connected if and only if it has only one point, so if you assume $T_1$, there are no nontrivial connected finite sets or spaces.

  • There are also connected $T_0$ spaces of any size you want (including any finite): if you take a nonempty set $X$ and fix $xin X$, then if you declare open sets to be exactly those containing $x$, then the resulting topology is $T_0$ and connected (all points except for $x$ are closed).

  • On any infinite set (possibly countable), the cofinite topology is $T_1$ and connected.

  • There are connected countably infinite Hausdorff spaces. (and even $T_{2frac{1}{2}}$-spaces).

  • A connected $T_4$ space with more than one point is necessarily uncountable. This follows easily from Urysohn's lemma (in fact, it follows that a nontrivial connected $T_4$ space has at least the cardinality of the continuum).

  • One can show that a countable $T_3$ space is $T_4$, so by the preceding point, a connected $T_3$ space cannot be countably infinite.


Incidentally, in topology, a continuum is a kind of connected set (namely one which is compact Hausdorff, or even metrisable, depending on the author).






share|cite|improve this answer











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    3 Answers
    3






    active

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    3 Answers
    3






    active

    oldest

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    active

    oldest

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    active

    oldest

    votes









    18












    $begingroup$

    First of all, $emptyset$ is connected. And so is every singleton.



    On the other hand, it is true that every connected metric space with more than one point is uncountable. But there are countable connected topological spaces.






    share|cite|improve this answer









    $endgroup$













    • $begingroup$
      I've heard some argue that the empty set is not connected, in a similar sense to 1 not being a prime number. See ncatlab.org/nlab/show/too+simple+to+be+simple for related thoughts.
      $endgroup$
      – Ben Millwood
      Feb 4 at 12:54










    • $begingroup$
      @BenMillwood, to me, the empty set is probably connected, but not $0$-connected (because $0$-connected implies $(-1)$-connected, and the latter is equivalent to being non-empty).
      $endgroup$
      – Carsten S
      Feb 4 at 14:14
















    18












    $begingroup$

    First of all, $emptyset$ is connected. And so is every singleton.



    On the other hand, it is true that every connected metric space with more than one point is uncountable. But there are countable connected topological spaces.






    share|cite|improve this answer









    $endgroup$













    • $begingroup$
      I've heard some argue that the empty set is not connected, in a similar sense to 1 not being a prime number. See ncatlab.org/nlab/show/too+simple+to+be+simple for related thoughts.
      $endgroup$
      – Ben Millwood
      Feb 4 at 12:54










    • $begingroup$
      @BenMillwood, to me, the empty set is probably connected, but not $0$-connected (because $0$-connected implies $(-1)$-connected, and the latter is equivalent to being non-empty).
      $endgroup$
      – Carsten S
      Feb 4 at 14:14














    18












    18








    18





    $begingroup$

    First of all, $emptyset$ is connected. And so is every singleton.



    On the other hand, it is true that every connected metric space with more than one point is uncountable. But there are countable connected topological spaces.






    share|cite|improve this answer









    $endgroup$



    First of all, $emptyset$ is connected. And so is every singleton.



    On the other hand, it is true that every connected metric space with more than one point is uncountable. But there are countable connected topological spaces.







    share|cite|improve this answer












    share|cite|improve this answer



    share|cite|improve this answer










    answered Feb 3 at 18:02









    José Carlos SantosJosé Carlos Santos

    169k23132237




    169k23132237












    • $begingroup$
      I've heard some argue that the empty set is not connected, in a similar sense to 1 not being a prime number. See ncatlab.org/nlab/show/too+simple+to+be+simple for related thoughts.
      $endgroup$
      – Ben Millwood
      Feb 4 at 12:54










    • $begingroup$
      @BenMillwood, to me, the empty set is probably connected, but not $0$-connected (because $0$-connected implies $(-1)$-connected, and the latter is equivalent to being non-empty).
      $endgroup$
      – Carsten S
      Feb 4 at 14:14


















    • $begingroup$
      I've heard some argue that the empty set is not connected, in a similar sense to 1 not being a prime number. See ncatlab.org/nlab/show/too+simple+to+be+simple for related thoughts.
      $endgroup$
      – Ben Millwood
      Feb 4 at 12:54










    • $begingroup$
      @BenMillwood, to me, the empty set is probably connected, but not $0$-connected (because $0$-connected implies $(-1)$-connected, and the latter is equivalent to being non-empty).
      $endgroup$
      – Carsten S
      Feb 4 at 14:14
















    $begingroup$
    I've heard some argue that the empty set is not connected, in a similar sense to 1 not being a prime number. See ncatlab.org/nlab/show/too+simple+to+be+simple for related thoughts.
    $endgroup$
    – Ben Millwood
    Feb 4 at 12:54




    $begingroup$
    I've heard some argue that the empty set is not connected, in a similar sense to 1 not being a prime number. See ncatlab.org/nlab/show/too+simple+to+be+simple for related thoughts.
    $endgroup$
    – Ben Millwood
    Feb 4 at 12:54












    $begingroup$
    @BenMillwood, to me, the empty set is probably connected, but not $0$-connected (because $0$-connected implies $(-1)$-connected, and the latter is equivalent to being non-empty).
    $endgroup$
    – Carsten S
    Feb 4 at 14:14




    $begingroup$
    @BenMillwood, to me, the empty set is probably connected, but not $0$-connected (because $0$-connected implies $(-1)$-connected, and the latter is equivalent to being non-empty).
    $endgroup$
    – Carsten S
    Feb 4 at 14:14











    10












    $begingroup$

    Consider the space $X = {a, b}$ equipped with the trivial topology (i.e. the only open sets are $X$ and $varnothing$). Then, the space is connected, but there are only a finite number of points in this example.






    share|cite|improve this answer









    $endgroup$









    • 9




      $begingroup$
      Even cooler is the pseudocircle. It's the smallest multiconnected space.
      $endgroup$
      – PyRulez
      Feb 3 at 23:10
















    10












    $begingroup$

    Consider the space $X = {a, b}$ equipped with the trivial topology (i.e. the only open sets are $X$ and $varnothing$). Then, the space is connected, but there are only a finite number of points in this example.






    share|cite|improve this answer









    $endgroup$









    • 9




      $begingroup$
      Even cooler is the pseudocircle. It's the smallest multiconnected space.
      $endgroup$
      – PyRulez
      Feb 3 at 23:10














    10












    10








    10





    $begingroup$

    Consider the space $X = {a, b}$ equipped with the trivial topology (i.e. the only open sets are $X$ and $varnothing$). Then, the space is connected, but there are only a finite number of points in this example.






    share|cite|improve this answer









    $endgroup$



    Consider the space $X = {a, b}$ equipped with the trivial topology (i.e. the only open sets are $X$ and $varnothing$). Then, the space is connected, but there are only a finite number of points in this example.







    share|cite|improve this answer












    share|cite|improve this answer



    share|cite|improve this answer










    answered Feb 3 at 18:01









    Kurtland ChuaKurtland Chua

    3541212




    3541212








    • 9




      $begingroup$
      Even cooler is the pseudocircle. It's the smallest multiconnected space.
      $endgroup$
      – PyRulez
      Feb 3 at 23:10














    • 9




      $begingroup$
      Even cooler is the pseudocircle. It's the smallest multiconnected space.
      $endgroup$
      – PyRulez
      Feb 3 at 23:10








    9




    9




    $begingroup$
    Even cooler is the pseudocircle. It's the smallest multiconnected space.
    $endgroup$
    – PyRulez
    Feb 3 at 23:10




    $begingroup$
    Even cooler is the pseudocircle. It's the smallest multiconnected space.
    $endgroup$
    – PyRulez
    Feb 3 at 23:10











    5












    $begingroup$

    Well, there's the empty set, as others have noted. Also, every singleton is connected. But you can say a bit more if you assume some separation.




    • Any finite $T_1$ space is discrete, so it is connected if and only if it has only one point, so if you assume $T_1$, there are no nontrivial connected finite sets or spaces.

    • There are also connected $T_0$ spaces of any size you want (including any finite): if you take a nonempty set $X$ and fix $xin X$, then if you declare open sets to be exactly those containing $x$, then the resulting topology is $T_0$ and connected (all points except for $x$ are closed).

    • On any infinite set (possibly countable), the cofinite topology is $T_1$ and connected.

    • There are connected countably infinite Hausdorff spaces. (and even $T_{2frac{1}{2}}$-spaces).

    • A connected $T_4$ space with more than one point is necessarily uncountable. This follows easily from Urysohn's lemma (in fact, it follows that a nontrivial connected $T_4$ space has at least the cardinality of the continuum).

    • One can show that a countable $T_3$ space is $T_4$, so by the preceding point, a connected $T_3$ space cannot be countably infinite.


    Incidentally, in topology, a continuum is a kind of connected set (namely one which is compact Hausdorff, or even metrisable, depending on the author).






    share|cite|improve this answer











    $endgroup$


















      5












      $begingroup$

      Well, there's the empty set, as others have noted. Also, every singleton is connected. But you can say a bit more if you assume some separation.




      • Any finite $T_1$ space is discrete, so it is connected if and only if it has only one point, so if you assume $T_1$, there are no nontrivial connected finite sets or spaces.

      • There are also connected $T_0$ spaces of any size you want (including any finite): if you take a nonempty set $X$ and fix $xin X$, then if you declare open sets to be exactly those containing $x$, then the resulting topology is $T_0$ and connected (all points except for $x$ are closed).

      • On any infinite set (possibly countable), the cofinite topology is $T_1$ and connected.

      • There are connected countably infinite Hausdorff spaces. (and even $T_{2frac{1}{2}}$-spaces).

      • A connected $T_4$ space with more than one point is necessarily uncountable. This follows easily from Urysohn's lemma (in fact, it follows that a nontrivial connected $T_4$ space has at least the cardinality of the continuum).

      • One can show that a countable $T_3$ space is $T_4$, so by the preceding point, a connected $T_3$ space cannot be countably infinite.


      Incidentally, in topology, a continuum is a kind of connected set (namely one which is compact Hausdorff, or even metrisable, depending on the author).






      share|cite|improve this answer











      $endgroup$
















        5












        5








        5





        $begingroup$

        Well, there's the empty set, as others have noted. Also, every singleton is connected. But you can say a bit more if you assume some separation.




        • Any finite $T_1$ space is discrete, so it is connected if and only if it has only one point, so if you assume $T_1$, there are no nontrivial connected finite sets or spaces.

        • There are also connected $T_0$ spaces of any size you want (including any finite): if you take a nonempty set $X$ and fix $xin X$, then if you declare open sets to be exactly those containing $x$, then the resulting topology is $T_0$ and connected (all points except for $x$ are closed).

        • On any infinite set (possibly countable), the cofinite topology is $T_1$ and connected.

        • There are connected countably infinite Hausdorff spaces. (and even $T_{2frac{1}{2}}$-spaces).

        • A connected $T_4$ space with more than one point is necessarily uncountable. This follows easily from Urysohn's lemma (in fact, it follows that a nontrivial connected $T_4$ space has at least the cardinality of the continuum).

        • One can show that a countable $T_3$ space is $T_4$, so by the preceding point, a connected $T_3$ space cannot be countably infinite.


        Incidentally, in topology, a continuum is a kind of connected set (namely one which is compact Hausdorff, or even metrisable, depending on the author).






        share|cite|improve this answer











        $endgroup$



        Well, there's the empty set, as others have noted. Also, every singleton is connected. But you can say a bit more if you assume some separation.




        • Any finite $T_1$ space is discrete, so it is connected if and only if it has only one point, so if you assume $T_1$, there are no nontrivial connected finite sets or spaces.

        • There are also connected $T_0$ spaces of any size you want (including any finite): if you take a nonempty set $X$ and fix $xin X$, then if you declare open sets to be exactly those containing $x$, then the resulting topology is $T_0$ and connected (all points except for $x$ are closed).

        • On any infinite set (possibly countable), the cofinite topology is $T_1$ and connected.

        • There are connected countably infinite Hausdorff spaces. (and even $T_{2frac{1}{2}}$-spaces).

        • A connected $T_4$ space with more than one point is necessarily uncountable. This follows easily from Urysohn's lemma (in fact, it follows that a nontrivial connected $T_4$ space has at least the cardinality of the continuum).

        • One can show that a countable $T_3$ space is $T_4$, so by the preceding point, a connected $T_3$ space cannot be countably infinite.


        Incidentally, in topology, a continuum is a kind of connected set (namely one which is compact Hausdorff, or even metrisable, depending on the author).







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Feb 4 at 1:39

























        answered Feb 4 at 1:19









        tomasztomasz

        23.9k23482




        23.9k23482






























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            0 1 What exactly does the error message beginning with E: mean? I assume it has to do with the file structure on Ubuntu-on-Windows, but what exactly? The following package was automatically installed and is no longer required: os-prober Use 'apt-get autoremove' to remove it. The following extra packages will be installed: libxslt1.1 The following NEW packages will be installed: libxslt1.1 xmlstarlet 0 upgraded, 2 newly installed, 0 to remove and 0 not upgraded. Need to get 435 kB of archives. After this operation, 1023 kB of additional disk space will be used. Get:1 http://archive.ubuntu.com/ubuntu/ trusty/main libxslt1.1 amd64 1.1.28-2build1 [145 kB] Get:2 http://archive.ubuntu.com/ubuntu/ trusty/universe xmlstarlet amd64 1.5.0-1 [290 kB] ...
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            張江高科駅

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            張江高科駅 ちょうこうこうか Zhangjiang Hi-Tech Park 所在地 中国上海市浦東新区 所属事業者 上海地鉄運営有限公司（上海軌道交通・駅詳細） 上海浦東現代有軌交通有限公司（駅詳細） テンプレートを表示 張江高科駅 各種表記 繁体字： 張江高科站 簡体字： 张江高科站 拼音 ： Zhàngjiāng Gāokē Zhàn 発音： ヂャンジャン ガオクー ヂャン 英文： Zhangjiang Hi-Tech Park Station テンプレートを表示 張江高科駅 （ちょうこうこうか-えき）は中華人民共和国上海市浦東新区に位置する、上海軌道交通2号線の駅である。 なお、張江高科駅前にある張江軌道電車の 張江地鉄駅駅 についてもここで述べる。 目次 1 利用可能な鉄道路線 2 沿革 3 駅構造 3.1 上海軌道交通 3.2 張江軌道電車 4 駅周辺 5 隣の駅 6 注釈 7 関連項目 利用可能な鉄道路線 上海軌道交通 ■ 2号線 上海浦東現代有軌交通有限公司 ■ 張江軌道電車 沿革 高架駅時代の姿 2000年12月27日 - 2号線の駅が高架駅として開業。 2010年 1月1日 - 張江軌道電車の開業に伴い、張江地鉄駅駅（停留所）を設置。 2月13日 - 駅地下化に伴い高架駅の運用停止。 2月24日 - 10日間の閉鎖を経て地下新駅の開業。 2017年2月9日 - リニューアル工事を経て再開業。乗降客の増加に対応し、ホームへのエレベーター、エスカレーターを増設、券売機エリアを拡張した [1] 。 駅構造 上海軌道交通 上海軌道交通 張江高科駅 5号口 ちょうこうこうか Zhangjiang Hi-Tech Park ◄ 竜陽路 (2.60km) (1.61km) 金科路 ► 所在地 中国上海市浦東新区松涛路 祖冲之路 （中国語版） 北緯31度12分11秒 東経121度34分50秒  /  北緯31.2...
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