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when are two topological spaces homeomorphic?

homeomorphic quotient spacesDemonstrate Topological Spaces are HomeomorphicHow to see that two sets are homeomorphic with respect to the subspace topologyIf two topological spaces have the same topological properties, are they homeomorphic?Showing that 2 topological spaces are homeomorphicHomeomorphic products of topological spacesShowing two topological spaces are not homeomorphicQuestion of whether two given spaces are homeomorphic.construct two non-homeomorphic topological spacesWhen are quotients of homeomorphic spaces homeomorphic?

3

$begingroup$

X and Y are topological spaces. If X is homeomorphic to a subspace of Y and Y is homeomorphic to
a subspace of X then X and Y are homeomorphic. Is this statement true ? justify

I know this is true but am unable to justify it. I tried to construct a homeomorphism but failed.

general-topology

share|cite|improve this question

asked Apr 30 at 18:33

Souvik DebSouvik Deb

225

$endgroup$

• 
  
  
  

  
  6
  

  
  
  
  
  
  

  
  $begingroup$
  What you know is false.
  $endgroup$
  – Moishe Kohan
  Apr 30 at 18:36
  

  
  
  
  

add a comment | 

3

$begingroup$

X and Y are topological spaces. If X is homeomorphic to a subspace of Y and Y is homeomorphic to
a subspace of X then X and Y are homeomorphic. Is this statement true ? justify

I know this is true but am unable to justify it. I tried to construct a homeomorphism but failed.

general-topology

share|cite|improve this question

asked Apr 30 at 18:33

Souvik DebSouvik Deb

225

$endgroup$

• 
  
  
  

  
  6
  

  
  
  
  
  
  

  
  $begingroup$
  What you know is false.
  $endgroup$
  – Moishe Kohan
  Apr 30 at 18:36
  

  
  
  
  

add a comment | 

$begingroup$

X and Y are topological spaces. If X is homeomorphic to a subspace of Y and Y is homeomorphic to
a subspace of X then X and Y are homeomorphic. Is this statement true ? justify

I know this is true but am unable to justify it. I tried to construct a homeomorphism but failed.

general-topology

share|cite|improve this question

asked Apr 30 at 18:33

Souvik DebSouvik Deb

225

$endgroup$

share|cite|improve this question

asked Apr 30 at 18:33

Souvik DebSouvik Deb

225

share|cite|improve this question

asked Apr 30 at 18:33

Souvik DebSouvik Deb

225

asked Apr 30 at 18:33

Souvik DebSouvik Deb

225

asked Apr 30 at 18:33

Souvik DebSouvik Deb

225

1 Answer
1

active

oldest

votes

11

$begingroup$

$mathbbR$ is homeomorphic to (0,1) which is a subspace of $[-1,2]$ and $[-1,2]$ is homeomorphic to $[-1,2]$ which is a subspace of $mathbbR$. But $[-1,2]$ is compact and $mathbbR$ is not so they are not homeomorphic.

share|cite|improve this answer

edited Apr 30 at 18:38

mihaild

2,409216

answered Apr 30 at 18:37

Adam ChalumeauAdam Chalumeau

1,064117

$endgroup$

add a comment | 

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11

$begingroup$

$mathbbR$ is homeomorphic to (0,1) which is a subspace of $[-1,2]$ and $[-1,2]$ is homeomorphic to $[-1,2]$ which is a subspace of $mathbbR$. But $[-1,2]$ is compact and $mathbbR$ is not so they are not homeomorphic.

share|cite|improve this answer

edited Apr 30 at 18:38

mihaild

2,409216

answered Apr 30 at 18:37

Adam ChalumeauAdam Chalumeau

1,064117

$endgroup$

add a comment | 

11

$begingroup$

$mathbbR$ is homeomorphic to (0,1) which is a subspace of $[-1,2]$ and $[-1,2]$ is homeomorphic to $[-1,2]$ which is a subspace of $mathbbR$. But $[-1,2]$ is compact and $mathbbR$ is not so they are not homeomorphic.

share|cite|improve this answer

edited Apr 30 at 18:38

mihaild

2,409216

answered Apr 30 at 18:37

Adam ChalumeauAdam Chalumeau

1,064117

$endgroup$

add a comment | 

$begingroup$

$mathbbR$ is homeomorphic to (0,1) which is a subspace of $[-1,2]$ and $[-1,2]$ is homeomorphic to $[-1,2]$ which is a subspace of $mathbbR$. But $[-1,2]$ is compact and $mathbbR$ is not so they are not homeomorphic.

share|cite|improve this answer

edited Apr 30 at 18:38

mihaild

2,409216

answered Apr 30 at 18:37

Adam ChalumeauAdam Chalumeau

1,064117

$endgroup$

share|cite|improve this answer

edited Apr 30 at 18:38

mihaild

2,409216

answered Apr 30 at 18:37

Adam ChalumeauAdam Chalumeau

1,064117

edited Apr 30 at 18:38

mihaild

2,409216

edited Apr 30 at 18:38

mihaild

2,409216

answered Apr 30 at 18:37

Adam ChalumeauAdam Chalumeau

1,064117

answered Apr 30 at 18:37

Adam ChalumeauAdam Chalumeau

1,064117