Fdtxjr

Deadlock Graph and Interpretation, solution to avoid

Realistic Alternatives to Dust: What Else Could Feed a Plankton Bloom?

How to make payment on the internet without leaving a money trail?

Protecting Dualbooting Windows from dangerous code (like rm -rf)

Does it makes sense to buy a new cycle to learn riding?

Does a dangling wire really electrocute me if I'm standing in water?

What is this 4-propeller plane?

How to create dashed lines/arrows in Illustrator

Does duplicating a spell with Wish count as casting that spell?

Is there a name of the flying bionic bird?

How long do I have to send my income tax payment to the IRS?

Should I use my personal or workplace e-mail when registering to external websites for work purpose?

Why is Grand Jury testimony secret?

Time travel alters history but people keep saying nothing's changed

What does Linus Torvalds mean when he says that Git "never ever" tracks a file?

What is the steepest angle that a canal can be traversable without locks?

Manuscript was "unsubmitted" because the manuscript was deposited in Arxiv Preprints

What does "rabbited" mean/imply in this sentence?

Inflated grade on resume at previous job, might former employer tell new employer?

How can I fix this gap between bookcases I made?

What could be the right powersource for 15 seconds lifespan disposable giant chainsaw?

I see my dog run

Access elements in std::string where positon of string is greater than its size

Flying Bloodthirsty Lampshades

Is $Bbb Q$ disconnected under general topology from $Bbb R$?

1

$begingroup$

I just completed the proof of that fact that $Bbb Q$ is disconnected. My question is how to find a separation of $Bbb Q$.

Let $X$ be a topological space. We say $A$ and $B$ forms a separation of $X$ if $Acap barB$ and $barAcap B$ are empty and $Acup B=X$.

Here $Bbb Q$ denotes the set of rational numbers.

Can someone please help with hints.

Thanks.

general-topology

share|cite|improve this question

edited Mar 23 at 8:19

Andrews

1,2812423

asked Mar 23 at 3:54

StammeringMathematicianStammeringMathematician

2,7951324

$endgroup$

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  I think you are missing some properties... “two subsets which are disjoint from each other[‘s] closure” is surely not enough, or else $0$ and $1$ would have given you an answer. Perhaps start by having the correct definition of “separated”.
  $endgroup$
  – Arturo Magidin
  Mar 23 at 3:55
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Please see the edit. @arturo
  $endgroup$
  – StammeringMathematician
  Mar 23 at 3:58
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  As a hint, the real numbers $mathbb R$ are connected and do not have the kind of "separation" you seek for $mathbb Q$. So the crux of the "separation" involves picking an irrational number (to split the rationals).
  $endgroup$
  – hardmath
  Mar 23 at 4:00
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  @hardmath I tried something like $(-infty,sqrt2)cup (sqrt2, infty)$
  $endgroup$
  – StammeringMathematician
  Mar 23 at 4:03
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Yes, that is the classic example!
  $endgroup$
  – hardmath
  Mar 23 at 4:05
  

  
  
  
  

 | 
show 3 more comments

1

$begingroup$

I just completed the proof of that fact that $Bbb Q$ is disconnected. My question is how to find a separation of $Bbb Q$.

Let $X$ be a topological space. We say $A$ and $B$ forms a separation of $X$ if $Acap barB$ and $barAcap B$ are empty and $Acup B=X$.

Here $Bbb Q$ denotes the set of rational numbers.

Can someone please help with hints.

Thanks.

general-topology

share|cite|improve this question

edited Mar 23 at 8:19

Andrews

1,2812423

asked Mar 23 at 3:54

StammeringMathematicianStammeringMathematician

2,7951324

$endgroup$

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  I think you are missing some properties... “two subsets which are disjoint from each other[‘s] closure” is surely not enough, or else $0$ and $1$ would have given you an answer. Perhaps start by having the correct definition of “separated”.
  $endgroup$
  – Arturo Magidin
  Mar 23 at 3:55
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Please see the edit. @arturo
  $endgroup$
  – StammeringMathematician
  Mar 23 at 3:58
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  As a hint, the real numbers $mathbb R$ are connected and do not have the kind of "separation" you seek for $mathbb Q$. So the crux of the "separation" involves picking an irrational number (to split the rationals).
  $endgroup$
  – hardmath
  Mar 23 at 4:00
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  @hardmath I tried something like $(-infty,sqrt2)cup (sqrt2, infty)$
  $endgroup$
  – StammeringMathematician
  Mar 23 at 4:03
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Yes, that is the classic example!
  $endgroup$
  – hardmath
  Mar 23 at 4:05
  

  
  
  
  

 | 
show 3 more comments

$begingroup$

I just completed the proof of that fact that $Bbb Q$ is disconnected. My question is how to find a separation of $Bbb Q$.

Let $X$ be a topological space. We say $A$ and $B$ forms a separation of $X$ if $Acap barB$ and $barAcap B$ are empty and $Acup B=X$.

Here $Bbb Q$ denotes the set of rational numbers.

Can someone please help with hints.

Thanks.

general-topology

share|cite|improve this question

edited Mar 23 at 8:19

Andrews

1,2812423

asked Mar 23 at 3:54

StammeringMathematicianStammeringMathematician

2,7951324

$endgroup$

share|cite|improve this question

edited Mar 23 at 8:19

Andrews

1,2812423

asked Mar 23 at 3:54

StammeringMathematicianStammeringMathematician

2,7951324

share|cite|improve this question

edited Mar 23 at 8:19

Andrews

1,2812423

asked Mar 23 at 3:54

StammeringMathematicianStammeringMathematician

2,7951324

edited Mar 23 at 8:19

Andrews

1,2812423

edited Mar 23 at 8:19

Andrews

1,2812423

asked Mar 23 at 3:54

StammeringMathematicianStammeringMathematician

2,7951324

asked Mar 23 at 3:54

StammeringMathematicianStammeringMathematician

2,7951324

1 Answer
1

active

oldest

votes

3

$begingroup$

$U= (-infty, sqrt2) cap mathbbQ$, is open in $mathbbQ$ as the intersection of an open subset of $mathbbR$ with $mathbbQ$ (definition of subspace topology) and similarly $V=(sqrt2, +infty) cap mathbbQ$ is also open in $mathbbQ$.

As $sqrt2 notin mathbbQ$ we have $mathbbQ = U cup V$ and $overlineU = U$ (closure in $mathbbQ$) and $overlineV=V$ and clearly $U cap V=emptyset$, so $U,V$ forms a separation of $mathbbQ$.

share|cite|improve this answer

answered Mar 23 at 5:50

Henno BrandsmaHenno Brandsma

116k349127

$endgroup$

add a comment | 

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

Sign up or log in

StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Post as a guest

Name

Email

Required, but never shown

StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3158932%2fis-bbb-q-disconnected-under-general-topology-from-bbb-r%23new-answer', 'question_page');

);

Post as a guest

Name

Email

Required, but never shown

3

$begingroup$

$U= (-infty, sqrt2) cap mathbbQ$, is open in $mathbbQ$ as the intersection of an open subset of $mathbbR$ with $mathbbQ$ (definition of subspace topology) and similarly $V=(sqrt2, +infty) cap mathbbQ$ is also open in $mathbbQ$.

As $sqrt2 notin mathbbQ$ we have $mathbbQ = U cup V$ and $overlineU = U$ (closure in $mathbbQ$) and $overlineV=V$ and clearly $U cap V=emptyset$, so $U,V$ forms a separation of $mathbbQ$.

share|cite|improve this answer

answered Mar 23 at 5:50

Henno BrandsmaHenno Brandsma

116k349127

$endgroup$

add a comment | 

3

$begingroup$

$U= (-infty, sqrt2) cap mathbbQ$, is open in $mathbbQ$ as the intersection of an open subset of $mathbbR$ with $mathbbQ$ (definition of subspace topology) and similarly $V=(sqrt2, +infty) cap mathbbQ$ is also open in $mathbbQ$.

As $sqrt2 notin mathbbQ$ we have $mathbbQ = U cup V$ and $overlineU = U$ (closure in $mathbbQ$) and $overlineV=V$ and clearly $U cap V=emptyset$, so $U,V$ forms a separation of $mathbbQ$.

share|cite|improve this answer

answered Mar 23 at 5:50

Henno BrandsmaHenno Brandsma

116k349127

$endgroup$

add a comment | 

$begingroup$

$U= (-infty, sqrt2) cap mathbbQ$, is open in $mathbbQ$ as the intersection of an open subset of $mathbbR$ with $mathbbQ$ (definition of subspace topology) and similarly $V=(sqrt2, +infty) cap mathbbQ$ is also open in $mathbbQ$.

As $sqrt2 notin mathbbQ$ we have $mathbbQ = U cup V$ and $overlineU = U$ (closure in $mathbbQ$) and $overlineV=V$ and clearly $U cap V=emptyset$, so $U,V$ forms a separation of $mathbbQ$.

share|cite|improve this answer

answered Mar 23 at 5:50

Henno BrandsmaHenno Brandsma

116k349127

$endgroup$

share|cite|improve this answer

answered Mar 23 at 5:50

Henno BrandsmaHenno Brandsma

116k349127

answered Mar 23 at 5:50

Henno BrandsmaHenno Brandsma

116k349127

answered Mar 23 at 5:50

Henno BrandsmaHenno Brandsma

116k349127