Topology for the set of all atlases on a topological manifold

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Is there a way to endow with a topology the set of all atlases of a given topological manifold $M$ ?

Which are the properties of this topology with respect to $M$ ? It would be very nice if this topology have paths between atlases which in some way correspond to isotopies between $(M,Phi)$ and $(M,Psi)$ where $Phi$ and $Psi$ are two atlases on $M$

Thanks.

general-topology manifolds

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edited Jul 20 at 22:42

asked Jul 20 at 19:19

Rodolfo Conde

395

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Could you give some examples of specific points, maybe paths in the suggested space?
  – Berci
  Jul 20 at 20:24
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  What is the purpose of topologizing the set of atlases?
  – Paul Frost
  Jul 20 at 22:20
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  I am working with manifolds and I have come to the point where I have a map from a topological space into the set of all atlases o a given topological manifold$M$. I would like this map to be continuous with this topology. I did a search on the net but I found nothing. I just wanted to know if there was some known topology on the set of all atlases on $M$
  – Rodolfo Conde
  Jul 20 at 22:27
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  What is that map?
  – Berci
  Jul 21 at 19:57
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  The map is of the form $fcolon X to mathcalA$ where $X$ is a topological space, mathcalA os the set of all atlases on $M$ and $f$ is surjective.
  – Rodolfo Conde
  Jul 26 at 5:59
  

  
  
  
  

add a comment | 

up vote
2
down vote

favorite

Is there a way to endow with a topology the set of all atlases of a given topological manifold $M$ ?

Which are the properties of this topology with respect to $M$ ? It would be very nice if this topology have paths between atlases which in some way correspond to isotopies between $(M,Phi)$ and $(M,Psi)$ where $Phi$ and $Psi$ are two atlases on $M$

Thanks.

general-topology manifolds

share|cite|improve this question

edited Jul 20 at 22:42

asked Jul 20 at 19:19

Rodolfo Conde

395

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Could you give some examples of specific points, maybe paths in the suggested space?
  – Berci
  Jul 20 at 20:24
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  What is the purpose of topologizing the set of atlases?
  – Paul Frost
  Jul 20 at 22:20
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  I am working with manifolds and I have come to the point where I have a map from a topological space into the set of all atlases o a given topological manifold$M$. I would like this map to be continuous with this topology. I did a search on the net but I found nothing. I just wanted to know if there was some known topology on the set of all atlases on $M$
  – Rodolfo Conde
  Jul 20 at 22:27
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  What is that map?
  – Berci
  Jul 21 at 19:57
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  The map is of the form $fcolon X to mathcalA$ where $X$ is a topological space, mathcalA os the set of all atlases on $M$ and $f$ is surjective.
  – Rodolfo Conde
  Jul 26 at 5:59
  

  
  
  
  

add a comment | 

up vote
2
down vote

favorite

up vote
2
down vote

favorite

Is there a way to endow with a topology the set of all atlases of a given topological manifold $M$ ?

Which are the properties of this topology with respect to $M$ ? It would be very nice if this topology have paths between atlases which in some way correspond to isotopies between $(M,Phi)$ and $(M,Psi)$ where $Phi$ and $Psi$ are two atlases on $M$

Thanks.

general-topology manifolds

share|cite|improve this question

edited Jul 20 at 22:42

asked Jul 20 at 19:19

Rodolfo Conde

395

Is there a way to endow with a topology the set of all atlases of a given topological manifold $M$ ?

Which are the properties of this topology with respect to $M$ ? It would be very nice if this topology have paths between atlases which in some way correspond to isotopies between $(M,Phi)$ and $(M,Psi)$ where $Phi$ and $Psi$ are two atlases on $M$

Thanks.

general-topology manifolds

share|cite|improve this question

edited Jul 20 at 22:42

asked Jul 20 at 19:19

Rodolfo Conde

395

share|cite|improve this question

share|cite|improve this question

edited Jul 20 at 22:42

edited Jul 20 at 22:42

edited Jul 20 at 22:42

asked Jul 20 at 19:19

Rodolfo Conde

395

asked Jul 20 at 19:19

Rodolfo Conde

395

asked Jul 20 at 19:19

Rodolfo Conde

395

395

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Could you give some examples of specific points, maybe paths in the suggested space?
  – Berci
  Jul 20 at 20:24
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  What is the purpose of topologizing the set of atlases?
  – Paul Frost
  Jul 20 at 22:20
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  I am working with manifolds and I have come to the point where I have a map from a topological space into the set of all atlases o a given topological manifold$M$. I would like this map to be continuous with this topology. I did a search on the net but I found nothing. I just wanted to know if there was some known topology on the set of all atlases on $M$
  – Rodolfo Conde
  Jul 20 at 22:27
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  What is that map?
  – Berci
  Jul 21 at 19:57
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  The map is of the form $fcolon X to mathcalA$ where $X$ is a topological space, mathcalA os the set of all atlases on $M$ and $f$ is surjective.
  – Rodolfo Conde
  Jul 26 at 5:59
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Could you give some examples of specific points, maybe paths in the suggested space?
  – Berci
  Jul 20 at 20:24
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  What is the purpose of topologizing the set of atlases?
  – Paul Frost
  Jul 20 at 22:20
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  I am working with manifolds and I have come to the point where I have a map from a topological space into the set of all atlases o a given topological manifold$M$. I would like this map to be continuous with this topology. I did a search on the net but I found nothing. I just wanted to know if there was some known topology on the set of all atlases on $M$
  – Rodolfo Conde
  Jul 20 at 22:27
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  What is that map?
  – Berci
  Jul 21 at 19:57
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  The map is of the form $fcolon X to mathcalA$ where $X$ is a topological space, mathcalA os the set of all atlases on $M$ and $f$ is surjective.
  – Rodolfo Conde
  Jul 26 at 5:59
  

  
  
  
  

1

1

Could you give some examples of specific points, maybe paths in the suggested space?
– Berci
Jul 20 at 20:24

Could you give some examples of specific points, maybe paths in the suggested space?
– Berci
Jul 20 at 20:24

What is the purpose of topologizing the set of atlases?
– Paul Frost
Jul 20 at 22:20

What is the purpose of topologizing the set of atlases?
– Paul Frost
Jul 20 at 22:20

I am working with manifolds and I have come to the point where I have a map from a topological space into the set of all atlases o a given topological manifold$M$. I would like this map to be continuous with this topology. I did a search on the net but I found nothing. I just wanted to know if there was some known topology on the set of all atlases on $M$
– Rodolfo Conde
Jul 20 at 22:27

I am working with manifolds and I have come to the point where I have a map from a topological space into the set of all atlases o a given topological manifold$M$. I would like this map to be continuous with this topology. I did a search on the net but I found nothing. I just wanted to know if there was some known topology on the set of all atlases on $M$
– Rodolfo Conde
Jul 20 at 22:27

What is that map?
– Berci
Jul 21 at 19:57

What is that map?
– Berci
Jul 21 at 19:57

The map is of the form $fcolon X to mathcalA$ where $X$ is a topological space, mathcalA os the set of all atlases on $M$ and $f$ is surjective.
– Rodolfo Conde
Jul 26 at 5:59

The map is of the form $fcolon X to mathcalA$ where $X$ is a topological space, mathcalA os the set of all atlases on $M$ and $f$ is surjective.
– Rodolfo Conde
Jul 26 at 5:59

add a comment | 

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