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Czes Kosniowski: Homeomorphic topological spaces $S^1times I / sim$ and $D^2$ [duplicate]

$(S^1 times [0,1])/sim$ homeomorphic to unit disk $ D^2$$(S^1 times [0,1])/sim$ homeomorphic to unit disk $ D^2$Proof that $mathbb R/[0,1]$ and $mathbb R$ are homeomorphicShow $(S^1times [0,1])/$~ is homeomorphic to $D^2$Topological , Homeomorphic version of $|S times S|=|S| $Why is $D^n/sim$ homeomorphic to $mathbbRP^n$?Why $(mathbb Qtimesmathbb Q)/(mathbb Ztimes=)$ is not homeomorphic to $(mathbb Q/mathbb Z)times(mathbb Q/=)$?Quotient spaces homeomorphic to the realsProve that $(Xtimes [0,1])/sim'$ is homeomorphic to $(X/sim)times[0,1].$RP2 is homeomorphic to $(I times I)/ sim$Homeomorphism between a quotient topology on $mathbbR^2$ and $mathbbR$

0

$begingroup$

This question already has an answer here:

• 
  $(S^1 times [0,1])/sim$ homeomorphic to unit disk $ D^2$
  
  1 answer
  
  

Let $X=S^1times I$ where $S^1subset mathbbC$ and $I=[0,1]subset mathbbR$ also define the equivalence relation ~ by $(x,t)$~$(y,s)$ if and only if $xt=ys$.

Prove that $X$/~ is homeomorphic to the unit disc $D^2=le1=xin mathbbC: $ with the induced topology.

Any hints would be appreciated.

general-topology algebraic-topology

share|cite|improve this question

edited yesterday

Paul Frost

11.6k3934

asked 2 days ago

AlfdavAlfdav

977

$endgroup$

marked as duplicate by Paul Frost, Jyrki Lahtonen, mrtaurho, Song, Ramiro yesterday

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
  

  
  
  
  
  
  

  
  $begingroup$
  what do you mean by $xt = ys$? Is $I$ also being considered a subset of $mathbbC$?
  $endgroup$
  – William
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Sorry $Isubset mathbbR$ i will fix it
  $endgroup$
  – Alfdav
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  Try imagining the equivalence relation on X. It identifies less points then you might expect because if $x neq y$ on the unit circle the only scalars you can multiply $x,y$ by to get them equal are 0 and 0. This construction is called the cone of $S^1$ which unsurprisingly is homeomorphic to a cone. So really what you want to show is that a cone is homeomorphic to a disc.
  $endgroup$
  – Connor Malin
  2 days ago
  

  
  
  
  

add a comment | 

0

$begingroup$

This question already has an answer here:

• 
  $(S^1 times [0,1])/sim$ homeomorphic to unit disk $ D^2$
  
  1 answer
  
  

Let $X=S^1times I$ where $S^1subset mathbbC$ and $I=[0,1]subset mathbbR$ also define the equivalence relation ~ by $(x,t)$~$(y,s)$ if and only if $xt=ys$.

Prove that $X$/~ is homeomorphic to the unit disc $D^2=le1=xin mathbbC: $ with the induced topology.

Any hints would be appreciated.

general-topology algebraic-topology

share|cite|improve this question

edited yesterday

Paul Frost

11.6k3934

asked 2 days ago

AlfdavAlfdav

977

$endgroup$

marked as duplicate by Paul Frost, Jyrki Lahtonen, mrtaurho, Song, Ramiro yesterday

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
  

  
  
  
  
  
  

  
  $begingroup$
  what do you mean by $xt = ys$? Is $I$ also being considered a subset of $mathbbC$?
  $endgroup$
  – William
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Sorry $Isubset mathbbR$ i will fix it
  $endgroup$
  – Alfdav
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  Try imagining the equivalence relation on X. It identifies less points then you might expect because if $x neq y$ on the unit circle the only scalars you can multiply $x,y$ by to get them equal are 0 and 0. This construction is called the cone of $S^1$ which unsurprisingly is homeomorphic to a cone. So really what you want to show is that a cone is homeomorphic to a disc.
  $endgroup$
  – Connor Malin
  2 days ago
  

  
  
  
  

add a comment | 

$begingroup$

This question already has an answer here:

• 
  $(S^1 times [0,1])/sim$ homeomorphic to unit disk $ D^2$
  
  1 answer
  
  

Let $X=S^1times I$ where $S^1subset mathbbC$ and $I=[0,1]subset mathbbR$ also define the equivalence relation ~ by $(x,t)$~$(y,s)$ if and only if $xt=ys$.

Prove that $X$/~ is homeomorphic to the unit disc $D^2=le1=xin mathbbC: $ with the induced topology.

Any hints would be appreciated.

general-topology algebraic-topology

share|cite|improve this question

edited yesterday

Paul Frost

11.6k3934

asked 2 days ago

AlfdavAlfdav

977

$endgroup$

share|cite|improve this question

edited yesterday

Paul Frost

11.6k3934

asked 2 days ago

AlfdavAlfdav

977

share|cite|improve this question

edited yesterday

Paul Frost

11.6k3934

asked 2 days ago

AlfdavAlfdav

977

edited yesterday

Paul Frost

11.6k3934

edited yesterday

Paul Frost

11.6k3934

asked 2 days ago

AlfdavAlfdav

977

asked 2 days ago

AlfdavAlfdav

977

1 Answer
1

active

oldest

votes

3

$begingroup$

Consider the function $fcolon S^1 times I to D^2$ defined by $f(x, t) = xt$. Show $f(x, t) = f(y, s)$ iff the pairs are equivalent, so $f$ induces a function on the quotient space

$$barfcolon (S^1 times I)/sim to D^2 $$

Now show that this function is a homeomorphism.

share|cite|improve this answer

edited 2 days ago

answered 2 days ago

WilliamWilliam

2,5851224

$endgroup$

add a comment | 

3

$begingroup$

Consider the function $fcolon S^1 times I to D^2$ defined by $f(x, t) = xt$. Show $f(x, t) = f(y, s)$ iff the pairs are equivalent, so $f$ induces a function on the quotient space

$$barfcolon (S^1 times I)/sim to D^2 $$

Now show that this function is a homeomorphism.

share|cite|improve this answer

edited 2 days ago

answered 2 days ago

WilliamWilliam

2,5851224

$endgroup$

add a comment | 

3

$begingroup$

Consider the function $fcolon S^1 times I to D^2$ defined by $f(x, t) = xt$. Show $f(x, t) = f(y, s)$ iff the pairs are equivalent, so $f$ induces a function on the quotient space

$$barfcolon (S^1 times I)/sim to D^2 $$

Now show that this function is a homeomorphism.

share|cite|improve this answer

edited 2 days ago

answered 2 days ago

WilliamWilliam

2,5851224

$endgroup$

add a comment | 

$begingroup$

Consider the function $fcolon S^1 times I to D^2$ defined by $f(x, t) = xt$. Show $f(x, t) = f(y, s)$ iff the pairs are equivalent, so $f$ induces a function on the quotient space

$$barfcolon (S^1 times I)/sim to D^2 $$

Now show that this function is a homeomorphism.

share|cite|improve this answer

edited 2 days ago

answered 2 days ago

WilliamWilliam

2,5851224

$endgroup$

share|cite|improve this answer

edited 2 days ago

answered 2 days ago

WilliamWilliam

2,5851224

answered 2 days ago

WilliamWilliam

2,5851224

answered 2 days ago

WilliamWilliam

2,5851224