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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            File:DeusFollowingSea.jpg

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            File:DeusFollowingSea.jpg From Wikipedia, the free encyclopedia Jump to navigation Jump to search File File history File usage No higher resolution available. DeusFollowingSea.jpg ‎ (238 × 211 pixels, file size: 12 KB, MIME type: image/jpeg ) Summary [ edit ] Media data and Non-free use rationale Description Cover art of the Deus CD Following Sea Author or copyright owner Deus Source (WP:NFCC#4) rocksucker.co.uk Date of publication 2012 Use in article (WP:NFCC#7) Following Sea Purpose of use in article (WP:NFCC#8) to serve as the primary means of visual identification at the top of the article dedicated to the work in question. Not replaceable with free media because (WP:NFCC#1) n.a. Minimal use (WP:NFCC#3) Low res image used in infobox of the article on the album Respect for commercial opportunities (WP:NFCC#2) n.a. Fair use Fair use of co
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