Is the Irrational Plane Path-Connected? [duplicate]If a topological space is a path-connected, it is also connected. The converse fails. Can this be remedied?Path homotopy in the planeQuestion on separations of path-connected spaces.There is no homeomorphic copy of $[0,1]$ in the plane which contains an open ballShowing that arcs do not separate the plane $mathbbR^2$Are the sets connected or path-connected?If $A subseteq mathbbR^n$ is path-connected, with path-connected complement, is $A$ necessarily simply-connected?Determine whether the following spaces are connected, totally disconnected or neitherPathological Continua which are Path Connected and Locally Path Connected.Is the product of path connected spaces also path connected in a topology other than the product topology?

Multi tool use
Multi tool use

What does the copyright in a dissertation protect exactly?

Picking a theme as a discovery writer

cd ` command meaning and how to exit it?

Gift for mentor after his thesis defense?

Splitting polygons and dividing attribute value proportionally using ArcGIS Pro?

How could a humanoid creature completely form within the span of 24 hours?

How does jetBlue determine its boarding order?

When does WordPress.org notify sites of new version?

Does every non-empty set admit an (affine) scheme structure (in ZFC)?

Is throwing dice a stochastic or a deterministic process?

How to increase row height of a table and vertically "align middle"?

Make me a minimum magic sum

Can a player choose to add detail and flavor to their character's spells and abilities?

Bash prompt takes only the first word of a hostname before the dot

How is it believable that Euron could so easily pull off this ambush?

Average of samples in a period of time

How does "politician" work as a job/career?

HTML folder located within IOS Image file?

Justification of physical currency in an interstellar civilization?

And now you see it

How do I give a darkroom course without negs from the attendees?

Would a legitimized Baratheon have the best claim for the Iron Throne?

Explaining intravenous drug abuse to a small child

Game artist computer workstation set-up – is this overkill?



Is the Irrational Plane Path-Connected? [duplicate]


If a topological space is a path-connected, it is also connected. The converse fails. Can this be remedied?Path homotopy in the planeQuestion on separations of path-connected spaces.There is no homeomorphic copy of $[0,1]$ in the plane which contains an open ballShowing that arcs do not separate the plane $mathbbR^2$Are the sets connected or path-connected?If $A subseteq mathbbR^n$ is path-connected, with path-connected complement, is $A$ necessarily simply-connected?Determine whether the following spaces are connected, totally disconnected or neitherPathological Continua which are Path Connected and Locally Path Connected.Is the product of path connected spaces also path connected in a topology other than the product topology?













1












$begingroup$



This question already has an answer here:



  • Is the Cartesian square of the set of irrational numbers path connected?

    1 answer



Is the set $I times I subset mathbbR^2$, where $I = mathbbRsetminus mathbbQ$ path connected?



It seems like it could be. Consider the path $gamma : (0,1) to mathbbR times mathbbR $ given by $gamma(t) = left(t, (1 - t^3)^frac13right)$. Fermat's last theorem guarantees $gamma((0,1)) subset I times I$. Thus, $gamma$ is a path of non-zero length contained in $I times I$. Perhaps paths like $gamma$ could be glued together to connect any two points of $I times I$?



Edit



As has been pointed out below, it is not the case that $gamma((0,1)) subset I times I$. Clearly $gamma(q) in mathbbQ times mathbbR$ when $q in mathbbQ$. Fermat's theorem only guarantees that $gamma((0,1)) subset mathbbR times I$. I had in mind the arc given implicitly by the equation $x^3 + y^3 = 1$ for $x,y in (0,1)$. While this arc will be a subset of $I times I$, it is in fact totally disconnected.










share|cite|improve this question











$endgroup$



marked as duplicate by Asaf Karagila general-topology
Users with the  general-topology badge can single-handedly close general-topology questions as duplicates and reopen them as needed.

StackExchange.ready(function()
if (StackExchange.options.isMobile) return;

$('.dupe-hammer-message-hover:not(.hover-bound)').each(function()
var $hover = $(this).addClass('hover-bound'),
$msg = $hover.siblings('.dupe-hammer-message');

$hover.hover(
function()
$hover.showInfoMessage('',
messageElement: $msg.clone().show(),
transient: false,
position: my: 'bottom left', at: 'top center', offsetTop: -7 ,
dismissable: false,
relativeToBody: true
);
,
function()
StackExchange.helpers.removeMessages();

);
);
);
2 hours ago


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.













  • 1




    $begingroup$
    Fermat's last theorem is a nice thought, but unfortunately $gamma((0,1)) nsubseteq I times I$, because $I times I$ is missing all points with rational $x$-coordinate, for instance.
    $endgroup$
    – Jane Doé
    3 hours ago











  • $begingroup$
    If $pi_1,pi_2$ are the projections onto the first and second coordinates, then $alpha_1=pi_1circ gamma$ and $alpha_2=pi_2circgamma$ are continuous. Since $(0,1)$ is connected then its image by $alpha_1$ and $alpha_2$ are connected. This mean that they contain only one point.
    $endgroup$
    – logarithm
    3 hours ago











  • $begingroup$
    I see now that the use of Fermat's last theorem was misguided. The arc composed of points $x, y in (0,1)$ such that $x^3 + y^3 = 1$ certainly lives in $I times I$, but is not connected. Meanwhile the arc I described contains points with rational coordinates.
    $endgroup$
    – Charles Hudgins
    3 hours ago















1












$begingroup$



This question already has an answer here:



  • Is the Cartesian square of the set of irrational numbers path connected?

    1 answer



Is the set $I times I subset mathbbR^2$, where $I = mathbbRsetminus mathbbQ$ path connected?



It seems like it could be. Consider the path $gamma : (0,1) to mathbbR times mathbbR $ given by $gamma(t) = left(t, (1 - t^3)^frac13right)$. Fermat's last theorem guarantees $gamma((0,1)) subset I times I$. Thus, $gamma$ is a path of non-zero length contained in $I times I$. Perhaps paths like $gamma$ could be glued together to connect any two points of $I times I$?



Edit



As has been pointed out below, it is not the case that $gamma((0,1)) subset I times I$. Clearly $gamma(q) in mathbbQ times mathbbR$ when $q in mathbbQ$. Fermat's theorem only guarantees that $gamma((0,1)) subset mathbbR times I$. I had in mind the arc given implicitly by the equation $x^3 + y^3 = 1$ for $x,y in (0,1)$. While this arc will be a subset of $I times I$, it is in fact totally disconnected.










share|cite|improve this question











$endgroup$



marked as duplicate by Asaf Karagila general-topology
Users with the  general-topology badge can single-handedly close general-topology questions as duplicates and reopen them as needed.

StackExchange.ready(function()
if (StackExchange.options.isMobile) return;

$('.dupe-hammer-message-hover:not(.hover-bound)').each(function()
var $hover = $(this).addClass('hover-bound'),
$msg = $hover.siblings('.dupe-hammer-message');

$hover.hover(
function()
$hover.showInfoMessage('',
messageElement: $msg.clone().show(),
transient: false,
position: my: 'bottom left', at: 'top center', offsetTop: -7 ,
dismissable: false,
relativeToBody: true
);
,
function()
StackExchange.helpers.removeMessages();

);
);
);
2 hours ago


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.













  • 1




    $begingroup$
    Fermat's last theorem is a nice thought, but unfortunately $gamma((0,1)) nsubseteq I times I$, because $I times I$ is missing all points with rational $x$-coordinate, for instance.
    $endgroup$
    – Jane Doé
    3 hours ago











  • $begingroup$
    If $pi_1,pi_2$ are the projections onto the first and second coordinates, then $alpha_1=pi_1circ gamma$ and $alpha_2=pi_2circgamma$ are continuous. Since $(0,1)$ is connected then its image by $alpha_1$ and $alpha_2$ are connected. This mean that they contain only one point.
    $endgroup$
    – logarithm
    3 hours ago











  • $begingroup$
    I see now that the use of Fermat's last theorem was misguided. The arc composed of points $x, y in (0,1)$ such that $x^3 + y^3 = 1$ certainly lives in $I times I$, but is not connected. Meanwhile the arc I described contains points with rational coordinates.
    $endgroup$
    – Charles Hudgins
    3 hours ago













1












1








1





$begingroup$



This question already has an answer here:



  • Is the Cartesian square of the set of irrational numbers path connected?

    1 answer



Is the set $I times I subset mathbbR^2$, where $I = mathbbRsetminus mathbbQ$ path connected?



It seems like it could be. Consider the path $gamma : (0,1) to mathbbR times mathbbR $ given by $gamma(t) = left(t, (1 - t^3)^frac13right)$. Fermat's last theorem guarantees $gamma((0,1)) subset I times I$. Thus, $gamma$ is a path of non-zero length contained in $I times I$. Perhaps paths like $gamma$ could be glued together to connect any two points of $I times I$?



Edit



As has been pointed out below, it is not the case that $gamma((0,1)) subset I times I$. Clearly $gamma(q) in mathbbQ times mathbbR$ when $q in mathbbQ$. Fermat's theorem only guarantees that $gamma((0,1)) subset mathbbR times I$. I had in mind the arc given implicitly by the equation $x^3 + y^3 = 1$ for $x,y in (0,1)$. While this arc will be a subset of $I times I$, it is in fact totally disconnected.










share|cite|improve this question











$endgroup$





This question already has an answer here:



  • Is the Cartesian square of the set of irrational numbers path connected?

    1 answer



Is the set $I times I subset mathbbR^2$, where $I = mathbbRsetminus mathbbQ$ path connected?



It seems like it could be. Consider the path $gamma : (0,1) to mathbbR times mathbbR $ given by $gamma(t) = left(t, (1 - t^3)^frac13right)$. Fermat's last theorem guarantees $gamma((0,1)) subset I times I$. Thus, $gamma$ is a path of non-zero length contained in $I times I$. Perhaps paths like $gamma$ could be glued together to connect any two points of $I times I$?



Edit



As has been pointed out below, it is not the case that $gamma((0,1)) subset I times I$. Clearly $gamma(q) in mathbbQ times mathbbR$ when $q in mathbbQ$. Fermat's theorem only guarantees that $gamma((0,1)) subset mathbbR times I$. I had in mind the arc given implicitly by the equation $x^3 + y^3 = 1$ for $x,y in (0,1)$. While this arc will be a subset of $I times I$, it is in fact totally disconnected.





This question already has an answer here:



  • Is the Cartesian square of the set of irrational numbers path connected?

    1 answer







general-topology analysis






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 3 hours ago







Charles Hudgins

















asked 3 hours ago









Charles HudginsCharles Hudgins

4357




4357




marked as duplicate by Asaf Karagila general-topology
Users with the  general-topology badge can single-handedly close general-topology questions as duplicates and reopen them as needed.

StackExchange.ready(function()
if (StackExchange.options.isMobile) return;

$('.dupe-hammer-message-hover:not(.hover-bound)').each(function()
var $hover = $(this).addClass('hover-bound'),
$msg = $hover.siblings('.dupe-hammer-message');

$hover.hover(
function()
$hover.showInfoMessage('',
messageElement: $msg.clone().show(),
transient: false,
position: my: 'bottom left', at: 'top center', offsetTop: -7 ,
dismissable: false,
relativeToBody: true
);
,
function()
StackExchange.helpers.removeMessages();

);
);
);
2 hours ago


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.









marked as duplicate by Asaf Karagila general-topology
Users with the  general-topology badge can single-handedly close general-topology questions as duplicates and reopen them as needed.

StackExchange.ready(function()
if (StackExchange.options.isMobile) return;

$('.dupe-hammer-message-hover:not(.hover-bound)').each(function()
var $hover = $(this).addClass('hover-bound'),
$msg = $hover.siblings('.dupe-hammer-message');

$hover.hover(
function()
$hover.showInfoMessage('',
messageElement: $msg.clone().show(),
transient: false,
position: my: 'bottom left', at: 'top center', offsetTop: -7 ,
dismissable: false,
relativeToBody: true
);
,
function()
StackExchange.helpers.removeMessages();

);
);
);
2 hours ago


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.









  • 1




    $begingroup$
    Fermat's last theorem is a nice thought, but unfortunately $gamma((0,1)) nsubseteq I times I$, because $I times I$ is missing all points with rational $x$-coordinate, for instance.
    $endgroup$
    – Jane Doé
    3 hours ago











  • $begingroup$
    If $pi_1,pi_2$ are the projections onto the first and second coordinates, then $alpha_1=pi_1circ gamma$ and $alpha_2=pi_2circgamma$ are continuous. Since $(0,1)$ is connected then its image by $alpha_1$ and $alpha_2$ are connected. This mean that they contain only one point.
    $endgroup$
    – logarithm
    3 hours ago











  • $begingroup$
    I see now that the use of Fermat's last theorem was misguided. The arc composed of points $x, y in (0,1)$ such that $x^3 + y^3 = 1$ certainly lives in $I times I$, but is not connected. Meanwhile the arc I described contains points with rational coordinates.
    $endgroup$
    – Charles Hudgins
    3 hours ago












  • 1




    $begingroup$
    Fermat's last theorem is a nice thought, but unfortunately $gamma((0,1)) nsubseteq I times I$, because $I times I$ is missing all points with rational $x$-coordinate, for instance.
    $endgroup$
    – Jane Doé
    3 hours ago











  • $begingroup$
    If $pi_1,pi_2$ are the projections onto the first and second coordinates, then $alpha_1=pi_1circ gamma$ and $alpha_2=pi_2circgamma$ are continuous. Since $(0,1)$ is connected then its image by $alpha_1$ and $alpha_2$ are connected. This mean that they contain only one point.
    $endgroup$
    – logarithm
    3 hours ago











  • $begingroup$
    I see now that the use of Fermat's last theorem was misguided. The arc composed of points $x, y in (0,1)$ such that $x^3 + y^3 = 1$ certainly lives in $I times I$, but is not connected. Meanwhile the arc I described contains points with rational coordinates.
    $endgroup$
    – Charles Hudgins
    3 hours ago







1




1




$begingroup$
Fermat's last theorem is a nice thought, but unfortunately $gamma((0,1)) nsubseteq I times I$, because $I times I$ is missing all points with rational $x$-coordinate, for instance.
$endgroup$
– Jane Doé
3 hours ago





$begingroup$
Fermat's last theorem is a nice thought, but unfortunately $gamma((0,1)) nsubseteq I times I$, because $I times I$ is missing all points with rational $x$-coordinate, for instance.
$endgroup$
– Jane Doé
3 hours ago













$begingroup$
If $pi_1,pi_2$ are the projections onto the first and second coordinates, then $alpha_1=pi_1circ gamma$ and $alpha_2=pi_2circgamma$ are continuous. Since $(0,1)$ is connected then its image by $alpha_1$ and $alpha_2$ are connected. This mean that they contain only one point.
$endgroup$
– logarithm
3 hours ago





$begingroup$
If $pi_1,pi_2$ are the projections onto the first and second coordinates, then $alpha_1=pi_1circ gamma$ and $alpha_2=pi_2circgamma$ are continuous. Since $(0,1)$ is connected then its image by $alpha_1$ and $alpha_2$ are connected. This mean that they contain only one point.
$endgroup$
– logarithm
3 hours ago













$begingroup$
I see now that the use of Fermat's last theorem was misguided. The arc composed of points $x, y in (0,1)$ such that $x^3 + y^3 = 1$ certainly lives in $I times I$, but is not connected. Meanwhile the arc I described contains points with rational coordinates.
$endgroup$
– Charles Hudgins
3 hours ago




$begingroup$
I see now that the use of Fermat's last theorem was misguided. The arc composed of points $x, y in (0,1)$ such that $x^3 + y^3 = 1$ certainly lives in $I times I$, but is not connected. Meanwhile the arc I described contains points with rational coordinates.
$endgroup$
– Charles Hudgins
3 hours ago










2 Answers
2






active

oldest

votes


















6












$begingroup$

$Itimes I$ is not path connected (or connected, for that matter), because the projection $pi(x,y)=x$ is a continuous map and $pi[Itimes I]$ is the disconnected set $I$.



In fact, in your candidate path, $gammaleft(frac12right)=left(frac12,fracsqrt[3]72right)notin Itimes I$.



On the other hand, $(Bbb RtimesBbb R)setminus(Bbb Qtimes Bbb Q)$ is path-connected, just like all complements of countable subsets of $Bbb R^2$ are.






share|cite|improve this answer











$endgroup$




















    2












    $begingroup$

    No. It's not even connected. Consider the map$$beginarrayrcccfcolon&Itimes I&longrightarrow&0,1\&(x,y)&mapsto&begincases1&text if y>frac12\0&text if y<frac12.endcasesendarray$$Then $f$ is continuous and $f(Itimes I)=0,1$, which is disconnected.






    share|cite|improve this answer









    $endgroup$



















      2 Answers
      2






      active

      oldest

      votes








      2 Answers
      2






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      6












      $begingroup$

      $Itimes I$ is not path connected (or connected, for that matter), because the projection $pi(x,y)=x$ is a continuous map and $pi[Itimes I]$ is the disconnected set $I$.



      In fact, in your candidate path, $gammaleft(frac12right)=left(frac12,fracsqrt[3]72right)notin Itimes I$.



      On the other hand, $(Bbb RtimesBbb R)setminus(Bbb Qtimes Bbb Q)$ is path-connected, just like all complements of countable subsets of $Bbb R^2$ are.






      share|cite|improve this answer











      $endgroup$

















        6












        $begingroup$

        $Itimes I$ is not path connected (or connected, for that matter), because the projection $pi(x,y)=x$ is a continuous map and $pi[Itimes I]$ is the disconnected set $I$.



        In fact, in your candidate path, $gammaleft(frac12right)=left(frac12,fracsqrt[3]72right)notin Itimes I$.



        On the other hand, $(Bbb RtimesBbb R)setminus(Bbb Qtimes Bbb Q)$ is path-connected, just like all complements of countable subsets of $Bbb R^2$ are.






        share|cite|improve this answer











        $endgroup$















          6












          6








          6





          $begingroup$

          $Itimes I$ is not path connected (or connected, for that matter), because the projection $pi(x,y)=x$ is a continuous map and $pi[Itimes I]$ is the disconnected set $I$.



          In fact, in your candidate path, $gammaleft(frac12right)=left(frac12,fracsqrt[3]72right)notin Itimes I$.



          On the other hand, $(Bbb RtimesBbb R)setminus(Bbb Qtimes Bbb Q)$ is path-connected, just like all complements of countable subsets of $Bbb R^2$ are.






          share|cite|improve this answer











          $endgroup$



          $Itimes I$ is not path connected (or connected, for that matter), because the projection $pi(x,y)=x$ is a continuous map and $pi[Itimes I]$ is the disconnected set $I$.



          In fact, in your candidate path, $gammaleft(frac12right)=left(frac12,fracsqrt[3]72right)notin Itimes I$.



          On the other hand, $(Bbb RtimesBbb R)setminus(Bbb Qtimes Bbb Q)$ is path-connected, just like all complements of countable subsets of $Bbb R^2$ are.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited 2 hours ago

























          answered 3 hours ago









          Saucy O'PathSaucy O'Path

          7,3891827




          7,3891827





















              2












              $begingroup$

              No. It's not even connected. Consider the map$$beginarrayrcccfcolon&Itimes I&longrightarrow&0,1\&(x,y)&mapsto&begincases1&text if y>frac12\0&text if y<frac12.endcasesendarray$$Then $f$ is continuous and $f(Itimes I)=0,1$, which is disconnected.






              share|cite|improve this answer









              $endgroup$

















                2












                $begingroup$

                No. It's not even connected. Consider the map$$beginarrayrcccfcolon&Itimes I&longrightarrow&0,1\&(x,y)&mapsto&begincases1&text if y>frac12\0&text if y<frac12.endcasesendarray$$Then $f$ is continuous and $f(Itimes I)=0,1$, which is disconnected.






                share|cite|improve this answer









                $endgroup$















                  2












                  2








                  2





                  $begingroup$

                  No. It's not even connected. Consider the map$$beginarrayrcccfcolon&Itimes I&longrightarrow&0,1\&(x,y)&mapsto&begincases1&text if y>frac12\0&text if y<frac12.endcasesendarray$$Then $f$ is continuous and $f(Itimes I)=0,1$, which is disconnected.






                  share|cite|improve this answer









                  $endgroup$



                  No. It's not even connected. Consider the map$$beginarrayrcccfcolon&Itimes I&longrightarrow&0,1\&(x,y)&mapsto&begincases1&text if y>frac12\0&text if y<frac12.endcasesendarray$$Then $f$ is continuous and $f(Itimes I)=0,1$, which is disconnected.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered 3 hours ago









                  José Carlos SantosJosé Carlos Santos

                  181k24142256




                  181k24142256













                      CB8axUYtdRyY52oZWt
                      n,PUjwJudLDFH,2 ObFXN

                      Popular posts from this blog

                      Log på Navigationsmenu

                      Siegen Nawigatsjuun

                      Nuuk Indholdsfortegnelse Etyomologi | Historie | Geografi | Transport og infrastruktur | Politik og administration | Uddannelsesinstitutioner | Kultur | Venskabsbyer | Noter | Eksterne henvisninger | Se også | Navigationsmenuwww.sermersooq.gl64°10′N 51°45′V / 64.167°N 51.750°V / 64.167; -51.75064°10′N 51°45′V / 64.167°N 51.750°V / 64.167; -51.750DMI - KlimanormalerSalmonsen, s. 850Grønlands Naturinstitut undersøger rensdyr i Akia og Maniitsoq foråret 2008Grønlands NaturinstitutNy vej til Qinngorput indviet i dagAntallet af biler i Nuuk må begrænsesNy taxacentral mødt med demonstrationKøreplan. Rute 1, 2 og 3SnescootersporNuukNord er for storSkoler i Kommuneqarfik SermersooqAtuarfik Samuel KleinschmidtKangillinguit AtuarfiatNuussuup AtuarfiaNuuk Internationale FriskoleIlinniarfissuaq, Grønlands SeminariumLedelseÅrsberetning for 2008Kunst og arkitekturÅrsberetning for 2008Julie om naturenNuuk KunstmuseumSilamiutGrønlands Nationalmuseum og ArkivStatistisk ÅrbogGrønlands LandsbibliotekStore koncerter på stribeVandhund nummer 1.000.000Kommuneqarfik Sermersooq – MalikForsidenVenskabsbyerLyngby-Taarbæk i GrønlandArctic Business NetworkWinter Cities 2008 i NuukDagligt opdaterede satellitbilleder fra NuukområdetKommuneqarfik Sermersooqs hjemmesideTurist i NuukGrønlands Statistiks databankGrønlands Hjemmestyres valgresultaterrrWorldCat124325457671310-5