What is the “natural” smooth structure on the subsets of $mathbbR^n$ that are topological manifolds? Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 23, 2019 at 23:30 UTC (7:30pm US/Eastern)Smooth structure on the topological spaceAn example of a smooth map between manifolds that is a topological embedding, but is NOT a smooth embedding.Smooth functions between manifolds and subsets of manifoldsAre topological manifolds with boundary metrizable?topological structure on smooth manifoldsTopological embedding and smooth structureHow can I see that the product smooth manifold structure is a maximal atlas?Let $θ$ be a smooth flow on an oriented smooth manifold $M$. Show that for every $tin mathbbR$, $θ_t: M to M$ is orientation-preserving.Definition of smooth manifolds: What is the difference between differentiable structures and smooth structures?Book for manifolds that are used in scientific computation
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What is the “natural” smooth structure on the subsets of $mathbbR^n$ that are topological manifolds?
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 23, 2019 at 23:30 UTC (7:30pm US/Eastern)Smooth structure on the topological spaceAn example of a smooth map between manifolds that is a topological embedding, but is NOT a smooth embedding.Smooth functions between manifolds and subsets of manifoldsAre topological manifolds with boundary metrizable?topological structure on smooth manifoldsTopological embedding and smooth structureHow can I see that the product smooth manifold structure is a maximal atlas?Let $θ$ be a smooth flow on an oriented smooth manifold $M$. Show that for every $tin mathbbR$, $θ_t: M to M$ is orientation-preserving.Definition of smooth manifolds: What is the difference between differentiable structures and smooth structures?Book for manifolds that are used in scientific computation
$begingroup$
Most of the differential topology book describe a manifold that is a subset of $mathbbR^n$ and without describing the smooth structure, they talk about their properties as smooth manifolds, such as a square which is mentioned in a question in Lee's book on p75.
What is the "natural" smooth structure on the subsets of $mathbbR^n$ that are topological manifolds ? Can you explicitly specify it ?
For example, if we were talking about the "natural" topology on the subsets of $mathbbR^n$, it would be the topology generated by the basis whose elements are the intersection of open balls with that subset; so now the topic is about smooth structure on those topological manifolds.
manifolds smooth-manifolds
asked Apr 2 at 8:01
onurcanbektasonurcanbektas
3,50211037
$endgroup$
add a comment |
$begingroup$
Most of the differential topology book describe a manifold that is a subset of $mathbbR^n$ and without describing the smooth structure, they talk about their properties as smooth manifolds, such as a square which is mentioned in a question in Lee's book on p75.
What is the "natural" smooth structure on the subsets of $mathbbR^n$ that are topological manifolds ? Can you explicitly specify it ?
For example, if we were talking about the "natural" topology on the subsets of $mathbbR^n$, it would be the topology generated by the basis whose elements are the intersection of open balls with that subset; so now the topic is about smooth structure on those topological manifolds.
manifolds smooth-manifolds
asked Apr 2 at 8:01
onurcanbektasonurcanbektas
3,50211037
$endgroup$
$begingroup$
Do you possibly mean open subsets of $mathbb R^n$?
$endgroup$
– lush
Apr 2 at 8:28
$begingroup$
@lush No. I'm asking specifially the closed subsets of $mathbbR^n$, such as square.
$endgroup$
– onurcanbektas
Apr 2 at 8:41
$begingroup$
@onurcanbektas: Note that the question in Lee's book doesn't treat the square as a smooth manifold or something which has smooth structure (although this is possible in the context of manifolds with corners). It asks about a diffeomorphism of the ambient space $mathbbR^2$ which maps a certain subset to another subset.
$endgroup$
– levap
Apr 2 at 13:11
$begingroup$
@levap To be able to diffeomorphism, do we not need spaces to be smooth manifolds ? After all, diffeomorphisms should preserve the smooth structure ?
$endgroup$
– onurcanbektas
Apr 2 at 18:20
$begingroup$
@onurcanbektas: You do, but the question asks to show that there is no diffeomorphism from $mathbbR^2$ to $mathbbR^2$ (which has a natural smooth structure) which maps a set $A subseteq mathbbR^2$ (circle) to a set $B subseteq mathbbR^2$ (boundary of a square). This is a legitimate question to ask for any two subsets of $mathbbR^2$, you don't need to endow them with a topology or a smooth structure.
$endgroup$
– levap
Apr 3 at 10:11
add a comment |
$begingroup$
Most of the differential topology book describe a manifold that is a subset of $mathbbR^n$ and without describing the smooth structure, they talk about their properties as smooth manifolds, such as a square which is mentioned in a question in Lee's book on p75.
What is the "natural" smooth structure on the subsets of $mathbbR^n$ that are topological manifolds ? Can you explicitly specify it ?
For example, if we were talking about the "natural" topology on the subsets of $mathbbR^n$, it would be the topology generated by the basis whose elements are the intersection of open balls with that subset; so now the topic is about smooth structure on those topological manifolds.
manifolds smooth-manifolds
asked Apr 2 at 8:01
onurcanbektasonurcanbektas
3,50211037
$endgroup$
Most of the differential topology book describe a manifold that is a subset of $mathbbR^n$ and without describing the smooth structure, they talk about their properties as smooth manifolds, such as a square which is mentioned in a question in Lee's book on p75.
What is the "natural" smooth structure on the subsets of $mathbbR^n$ that are topological manifolds ? Can you explicitly specify it ?
For example, if we were talking about the "natural" topology on the subsets of $mathbbR^n$, it would be the topology generated by the basis whose elements are the intersection of open balls with that subset; so now the topic is about smooth structure on those topological manifolds.
manifolds smooth-manifolds
manifolds smooth-manifolds
asked Apr 2 at 8:01
onurcanbektasonurcanbektas
3,50211037
asked Apr 2 at 8:01
onurcanbektasonurcanbektas
3,50211037
asked Apr 2 at 8:01
onurcanbektasonurcanbektas
3,50211037
asked Apr 2 at 8:01
onurcanbektasonurcanbektas
3,50211037
asked Apr 2 at 8:01
onurcanbektasonurcanbektas
3,50211037
3,50211037
$begingroup$
Do you possibly mean open subsets of $mathbb R^n$?
$endgroup$
– lush
Apr 2 at 8:28
$begingroup$
@lush No. I'm asking specifially the closed subsets of $mathbbR^n$, such as square.
$endgroup$
– onurcanbektas
Apr 2 at 8:41
$begingroup$
@onurcanbektas: Note that the question in Lee's book doesn't treat the square as a smooth manifold or something which has smooth structure (although this is possible in the context of manifolds with corners). It asks about a diffeomorphism of the ambient space $mathbbR^2$ which maps a certain subset to another subset.
$endgroup$
– levap
Apr 2 at 13:11
$begingroup$
@levap To be able to diffeomorphism, do we not need spaces to be smooth manifolds ? After all, diffeomorphisms should preserve the smooth structure ?
$endgroup$
– onurcanbektas
Apr 2 at 18:20
$begingroup$
@onurcanbektas: You do, but the question asks to show that there is no diffeomorphism from $mathbbR^2$ to $mathbbR^2$ (which has a natural smooth structure) which maps a set $A subseteq mathbbR^2$ (circle) to a set $B subseteq mathbbR^2$ (boundary of a square). This is a legitimate question to ask for any two subsets of $mathbbR^2$, you don't need to endow them with a topology or a smooth structure.
$endgroup$
– levap
Apr 3 at 10:11
add a comment |
$begingroup$
Do you possibly mean open subsets of $mathbb R^n$?
$endgroup$
– lush
Apr 2 at 8:28
$begingroup$
@lush No. I'm asking specifially the closed subsets of $mathbbR^n$, such as square.
$endgroup$
– onurcanbektas
Apr 2 at 8:41
$begingroup$
@onurcanbektas: Note that the question in Lee's book doesn't treat the square as a smooth manifold or something which has smooth structure (although this is possible in the context of manifolds with corners). It asks about a diffeomorphism of the ambient space $mathbbR^2$ which maps a certain subset to another subset.
$endgroup$
– levap
Apr 2 at 13:11
$begingroup$
@levap To be able to diffeomorphism, do we not need spaces to be smooth manifolds ? After all, diffeomorphisms should preserve the smooth structure ?
$endgroup$
– onurcanbektas
Apr 2 at 18:20
$begingroup$
@onurcanbektas: You do, but the question asks to show that there is no diffeomorphism from $mathbbR^2$ to $mathbbR^2$ (which has a natural smooth structure) which maps a set $A subseteq mathbbR^2$ (circle) to a set $B subseteq mathbbR^2$ (boundary of a square). This is a legitimate question to ask for any two subsets of $mathbbR^2$, you don't need to endow them with a topology or a smooth structure.
$endgroup$
– levap
Apr 3 at 10:11
$begingroup$
Do you possibly mean open subsets of $mathbb R^n$?
$endgroup$
– lush
Apr 2 at 8:28
$begingroup$
Do you possibly mean open subsets of $mathbb R^n$?
$endgroup$
– lush
Apr 2 at 8:28
$begingroup$
@lush No. I'm asking specifially the closed subsets of $mathbbR^n$, such as square.
$endgroup$
– onurcanbektas
Apr 2 at 8:41
$begingroup$
@lush No. I'm asking specifially the closed subsets of $mathbbR^n$, such as square.
$endgroup$
– onurcanbektas
Apr 2 at 8:41
$begingroup$
@onurcanbektas: Note that the question in Lee's book doesn't treat the square as a smooth manifold or something which has smooth structure (although this is possible in the context of manifolds with corners). It asks about a diffeomorphism of the ambient space $mathbbR^2$ which maps a certain subset to another subset.
$endgroup$
– levap
Apr 2 at 13:11
$begingroup$
@onurcanbektas: Note that the question in Lee's book doesn't treat the square as a smooth manifold or something which has smooth structure (although this is possible in the context of manifolds with corners). It asks about a diffeomorphism of the ambient space $mathbbR^2$ which maps a certain subset to another subset.
$endgroup$
– levap
Apr 2 at 13:11
$begingroup$
@levap To be able to diffeomorphism, do we not need spaces to be smooth manifolds ? After all, diffeomorphisms should preserve the smooth structure ?
$endgroup$
– onurcanbektas
Apr 2 at 18:20
$begingroup$
@levap To be able to diffeomorphism, do we not need spaces to be smooth manifolds ? After all, diffeomorphisms should preserve the smooth structure ?
$endgroup$
– onurcanbektas
Apr 2 at 18:20
$begingroup$
@onurcanbektas: You do, but the question asks to show that there is no diffeomorphism from $mathbbR^2$ to $mathbbR^2$ (which has a natural smooth structure) which maps a set $A subseteq mathbbR^2$ (circle) to a set $B subseteq mathbbR^2$ (boundary of a square). This is a legitimate question to ask for any two subsets of $mathbbR^2$, you don't need to endow them with a topology or a smooth structure.
$endgroup$
– levap
Apr 3 at 10:11
$begingroup$
@onurcanbektas: You do, but the question asks to show that there is no diffeomorphism from $mathbbR^2$ to $mathbbR^2$ (which has a natural smooth structure) which maps a set $A subseteq mathbbR^2$ (circle) to a set $B subseteq mathbbR^2$ (boundary of a square). This is a legitimate question to ask for any two subsets of $mathbbR^2$, you don't need to endow them with a topology or a smooth structure.
$endgroup$
– levap
Apr 3 at 10:11
add a comment |
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$begingroup$
Do you possibly mean open subsets of $mathbb R^n$?
$endgroup$
– lush
Apr 2 at 8:28
$begingroup$
@lush No. I'm asking specifially the closed subsets of $mathbbR^n$, such as square.
$endgroup$
– onurcanbektas
Apr 2 at 8:41
$begingroup$
@onurcanbektas: Note that the question in Lee's book doesn't treat the square as a smooth manifold or something which has smooth structure (although this is possible in the context of manifolds with corners). It asks about a diffeomorphism of the ambient space $mathbbR^2$ which maps a certain subset to another subset.
$endgroup$
– levap
Apr 2 at 13:11
$begingroup$
@levap To be able to diffeomorphism, do we not need spaces to be smooth manifolds ? After all, diffeomorphisms should preserve the smooth structure ?
$endgroup$
– onurcanbektas
Apr 2 at 18:20
$begingroup$
@onurcanbektas: You do, but the question asks to show that there is no diffeomorphism from $mathbbR^2$ to $mathbbR^2$ (which has a natural smooth structure) which maps a set $A subseteq mathbbR^2$ (circle) to a set $B subseteq mathbbR^2$ (boundary of a square). This is a legitimate question to ask for any two subsets of $mathbbR^2$, you don't need to endow them with a topology or a smooth structure.
$endgroup$
– levap
Apr 3 at 10:11