Confusion about the definition of an n-differentiable manifold in Bredon's book “Topology and Geometry”
In Definition 2.1 on page 68 and 69 Glen E. Bredon defines a $n$-differentiable manifold as a second countable Hausdorff space $M^n$ and a collection of charts such that:
a chart is a homeomorphism $phi:Uto U'subsetmathbb{R}^n$ where $U$ is open in $M^n$ and $U'$ is open in $mathbb{R}^n$;
each point $xin M$ is in the domain of some chart;
- for charts $phi:Uto U'subset mathbb{R}^n$ and $psi:Vto V'subsetmathbb{R}^n$ we have that the “change of coordinates” $phipsi^{-1}:psi(Ucap V)tophi(Ucap V)$ is $C^infty$; and
- the collection of charts is maximal with properties 1, 2, and 3.
I'm confused about property 2 where he talks about a point $xin M$ being element of an open subset of $M^n$.
In definition 2.4 on the next two pages he talks about an functionally structured Hausdorff space $(F,M^n)$ where he also talks about points $xin M$ being elements of open subsets of $M^n$.
My guess is that he writes $M^n$ to imply that he talks about an $n$-differentiable manifold or is he talking about an embedding of $M$ in $M^n$?
differential-topology smooth-manifolds
asked Jun 12 at 9:43
messias
61
add a comment |
In Definition 2.1 on page 68 and 69 Glen E. Bredon defines a $n$-differentiable manifold as a second countable Hausdorff space $M^n$ and a collection of charts such that:
a chart is a homeomorphism $phi:Uto U'subsetmathbb{R}^n$ where $U$ is open in $M^n$ and $U'$ is open in $mathbb{R}^n$;
each point $xin M$ is in the domain of some chart;
- for charts $phi:Uto U'subset mathbb{R}^n$ and $psi:Vto V'subsetmathbb{R}^n$ we have that the “change of coordinates” $phipsi^{-1}:psi(Ucap V)tophi(Ucap V)$ is $C^infty$; and
- the collection of charts is maximal with properties 1, 2, and 3.
I'm confused about property 2 where he talks about a point $xin M$ being element of an open subset of $M^n$.
In definition 2.4 on the next two pages he talks about an functionally structured Hausdorff space $(F,M^n)$ where he also talks about points $xin M$ being elements of open subsets of $M^n$.
My guess is that he writes $M^n$ to imply that he talks about an $n$-differentiable manifold or is he talking about an embedding of $M$ in $M^n$?
differential-topology smooth-manifolds
asked Jun 12 at 9:43
messias
61
1
The notation $M^n$ is usually used to indicate that $M$ is of dimension $n$, indeed
– Max
Jun 12 at 9:57
add a comment |
In Definition 2.1 on page 68 and 69 Glen E. Bredon defines a $n$-differentiable manifold as a second countable Hausdorff space $M^n$ and a collection of charts such that:
a chart is a homeomorphism $phi:Uto U'subsetmathbb{R}^n$ where $U$ is open in $M^n$ and $U'$ is open in $mathbb{R}^n$;
each point $xin M$ is in the domain of some chart;
- for charts $phi:Uto U'subset mathbb{R}^n$ and $psi:Vto V'subsetmathbb{R}^n$ we have that the “change of coordinates” $phipsi^{-1}:psi(Ucap V)tophi(Ucap V)$ is $C^infty$; and
- the collection of charts is maximal with properties 1, 2, and 3.
I'm confused about property 2 where he talks about a point $xin M$ being element of an open subset of $M^n$.
In definition 2.4 on the next two pages he talks about an functionally structured Hausdorff space $(F,M^n)$ where he also talks about points $xin M$ being elements of open subsets of $M^n$.
My guess is that he writes $M^n$ to imply that he talks about an $n$-differentiable manifold or is he talking about an embedding of $M$ in $M^n$?
differential-topology smooth-manifolds
asked Jun 12 at 9:43
messias
61
In Definition 2.1 on page 68 and 69 Glen E. Bredon defines a $n$-differentiable manifold as a second countable Hausdorff space $M^n$ and a collection of charts such that:
a chart is a homeomorphism $phi:Uto U'subsetmathbb{R}^n$ where $U$ is open in $M^n$ and $U'$ is open in $mathbb{R}^n$;
each point $xin M$ is in the domain of some chart;
- for charts $phi:Uto U'subset mathbb{R}^n$ and $psi:Vto V'subsetmathbb{R}^n$ we have that the “change of coordinates” $phipsi^{-1}:psi(Ucap V)tophi(Ucap V)$ is $C^infty$; and
- the collection of charts is maximal with properties 1, 2, and 3.
I'm confused about property 2 where he talks about a point $xin M$ being element of an open subset of $M^n$.
In definition 2.4 on the next two pages he talks about an functionally structured Hausdorff space $(F,M^n)$ where he also talks about points $xin M$ being elements of open subsets of $M^n$.
My guess is that he writes $M^n$ to imply that he talks about an $n$-differentiable manifold or is he talking about an embedding of $M$ in $M^n$?
differential-topology smooth-manifolds
differential-topology smooth-manifolds
asked Jun 12 at 9:43
messias
61
asked Jun 12 at 9:43
messias
61
asked Jun 12 at 9:43
messias
61
asked Jun 12 at 9:43
messias
61
asked Jun 12 at 9:43
messias
61
61
1
The notation $M^n$ is usually used to indicate that $M$ is of dimension $n$, indeed
– Max
Jun 12 at 9:57
add a comment |
1
The notation $M^n$ is usually used to indicate that $M$ is of dimension $n$, indeed
– Max
Jun 12 at 9:57
1
1
The notation $M^n$ is usually used to indicate that $M$ is of dimension $n$, indeed
– Max
Jun 12 at 9:57
The notation $M^n$ is usually used to indicate that $M$ is of dimension $n$, indeed
– Max
Jun 12 at 9:57
add a comment |
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From the comment above by @Max.
You are right, the notation $M^n$ is used to indicate that $M$ is an $n$-dimensional differentiable manifold. It is not used in the sense of the Cartesian product $M times dots times M$.
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From the comment above by @Max.
You are right, the notation $M^n$ is used to indicate that $M$ is an $n$-dimensional differentiable manifold. It is not used in the sense of the Cartesian product $M times dots times M$.
add a comment |
From the comment above by @Max.
You are right, the notation $M^n$ is used to indicate that $M$ is an $n$-dimensional differentiable manifold. It is not used in the sense of the Cartesian product $M times dots times M$.
add a comment |
From the comment above by @Max.
You are right, the notation $M^n$ is used to indicate that $M$ is an $n$-dimensional differentiable manifold. It is not used in the sense of the Cartesian product $M times dots times M$.
From the comment above by @Max.
You are right, the notation $M^n$ is used to indicate that $M$ is an $n$-dimensional differentiable manifold. It is not used in the sense of the Cartesian product $M times dots times M$.
answered Dec 8 at 16:12
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Brahadeesh
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The notation $M^n$ is usually used to indicate that $M$ is of dimension $n$, indeed
– Max
Jun 12 at 9:57