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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 $D^2=xin mathbbR^2: =x$ with the induced topology.

Any hints would be appreciated.

general-topology algebraic-topology

share|cite|improve this question

edited 2 days ago

Paul Frost

11.6k3934

asked Mar 9 at 21:06

AlfdavAlfdav

1027

$endgroup$

marked as duplicate by Paul Frost, Jyrki Lahtonen, mrtaurho, Song, Ramiro 2 days 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
  

  
  
  
  
  
  

  
  $begingroup$
  what do you mean by $xt = ys$? Is $I$ also being considered a subset of $mathbbC$?
  $endgroup$
  – William
  Mar 9 at 21:12
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Sorry $Isubset mathbbR$ i will fix it
  $endgroup$
  – Alfdav
  Mar 9 at 21:21
  

  
  
  
  


• 
  
  
  

  
  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
  Mar 9 at 21:38
  

  
  
  
  

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 $D^2=xin mathbbR^2: =x$ with the induced topology.

Any hints would be appreciated.

general-topology algebraic-topology

share|cite|improve this question

edited 2 days ago

Paul Frost

11.6k3934

asked Mar 9 at 21:06

AlfdavAlfdav

1027

$endgroup$

marked as duplicate by Paul Frost, Jyrki Lahtonen, mrtaurho, Song, Ramiro 2 days 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
  

  
  
  
  
  
  

  
  $begingroup$
  what do you mean by $xt = ys$? Is $I$ also being considered a subset of $mathbbC$?
  $endgroup$
  – William
  Mar 9 at 21:12
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Sorry $Isubset mathbbR$ i will fix it
  $endgroup$
  – Alfdav
  Mar 9 at 21:21
  

  
  
  
  


• 
  
  
  

  
  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
  Mar 9 at 21:38
  

  
  
  
  

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 $D^2=xin mathbbR^2: =x$ with the induced topology.

Any hints would be appreciated.

general-topology algebraic-topology

share|cite|improve this question

edited 2 days ago

Paul Frost

11.6k3934

asked Mar 9 at 21:06

AlfdavAlfdav

1027

$endgroup$

share|cite|improve this question

edited 2 days ago

Paul Frost

11.6k3934

asked Mar 9 at 21:06

AlfdavAlfdav

1027

share|cite|improve this question

edited 2 days ago

Paul Frost

11.6k3934

asked Mar 9 at 21:06

AlfdavAlfdav

1027

edited 2 days ago

Paul Frost

11.6k3934

edited 2 days ago

Paul Frost

11.6k3934

asked Mar 9 at 21:06

AlfdavAlfdav

1027

asked Mar 9 at 21:06

AlfdavAlfdav

1027

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 Mar 9 at 21:52

answered Mar 9 at 21:45

WilliamWilliam

2,6051224

$endgroup$

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 Mar 9 at 21:52

answered Mar 9 at 21:45

WilliamWilliam

2,6051224

$endgroup$

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 Mar 9 at 21:52

answered Mar 9 at 21:45

WilliamWilliam

2,6051224

$endgroup$

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 Mar 9 at 21:52

answered Mar 9 at 21:45

WilliamWilliam

2,6051224

$endgroup$

share|cite|improve this answer

edited Mar 9 at 21:52

answered Mar 9 at 21:45

WilliamWilliam

2,6051224

answered Mar 9 at 21:45

WilliamWilliam

2,6051224

answered Mar 9 at 21:45

WilliamWilliam

2,6051224