Does a topological submanifold necessarily carry the subspace topology? Is it locally closed?
This answer suggests the following definition of a topological submanifold.
Definition. Let $N$ be an $n$-dimensional topological manifold. A subset $Msubset N$ is called an $m$-dimensional topological submanifold if for every $xin M$ there exists an (open) neighborhood $U$ of $x$ in $N$ and a diffeomorphism $phi: Uto Vsubset R^n$ ($V$ is open) such that $phi(Mcap U)= Lcap V$, where $L$ is an $m$-dimensional linear subspace in $R^n$.
In other words, this means the restriction of an atlas of $N$ gives an atlas of $M$ w.r.t a linear subspace.
- Is $Msubset N$ locally closed?
- Now suppose we fix some topology on $M$ such that the inclusion $iota:Mhookrightarrow N$ is continuous. Must this be the subspace topology?
In other words, does an "immersion" of topological manifolds have to be a topological embedding?
manifolds
asked Nov 29 '18 at 12:21
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This answer suggests the following definition of a topological submanifold.
Definition. Let $N$ be an $n$-dimensional topological manifold. A subset $Msubset N$ is called an $m$-dimensional topological submanifold if for every $xin M$ there exists an (open) neighborhood $U$ of $x$ in $N$ and a diffeomorphism $phi: Uto Vsubset R^n$ ($V$ is open) such that $phi(Mcap U)= Lcap V$, where $L$ is an $m$-dimensional linear subspace in $R^n$.
In other words, this means the restriction of an atlas of $N$ gives an atlas of $M$ w.r.t a linear subspace.
- Is $Msubset N$ locally closed?
- Now suppose we fix some topology on $M$ such that the inclusion $iota:Mhookrightarrow N$ is continuous. Must this be the subspace topology?
In other words, does an "immersion" of topological manifolds have to be a topological embedding?
manifolds
asked Nov 29 '18 at 12:21
Arrow
5,07621445
add a comment |
This answer suggests the following definition of a topological submanifold.
Definition. Let $N$ be an $n$-dimensional topological manifold. A subset $Msubset N$ is called an $m$-dimensional topological submanifold if for every $xin M$ there exists an (open) neighborhood $U$ of $x$ in $N$ and a diffeomorphism $phi: Uto Vsubset R^n$ ($V$ is open) such that $phi(Mcap U)= Lcap V$, where $L$ is an $m$-dimensional linear subspace in $R^n$.
In other words, this means the restriction of an atlas of $N$ gives an atlas of $M$ w.r.t a linear subspace.
- Is $Msubset N$ locally closed?
- Now suppose we fix some topology on $M$ such that the inclusion $iota:Mhookrightarrow N$ is continuous. Must this be the subspace topology?
In other words, does an "immersion" of topological manifolds have to be a topological embedding?
manifolds
asked Nov 29 '18 at 12:21
Arrow
5,07621445
This answer suggests the following definition of a topological submanifold.
Definition. Let $N$ be an $n$-dimensional topological manifold. A subset $Msubset N$ is called an $m$-dimensional topological submanifold if for every $xin M$ there exists an (open) neighborhood $U$ of $x$ in $N$ and a diffeomorphism $phi: Uto Vsubset R^n$ ($V$ is open) such that $phi(Mcap U)= Lcap V$, where $L$ is an $m$-dimensional linear subspace in $R^n$.
In other words, this means the restriction of an atlas of $N$ gives an atlas of $M$ w.r.t a linear subspace.
- Is $Msubset N$ locally closed?
- Now suppose we fix some topology on $M$ such that the inclusion $iota:Mhookrightarrow N$ is continuous. Must this be the subspace topology?
In other words, does an "immersion" of topological manifolds have to be a topological embedding?
manifolds
manifolds
asked Nov 29 '18 at 12:21
Arrow
5,07621445
asked Nov 29 '18 at 12:21
Arrow
5,07621445
edited Nov 29 '18 at 12:30
asked Nov 29 '18 at 12:21
Arrow
5,07621445
asked Nov 29 '18 at 12:21
Arrow
5,07621445
asked Nov 29 '18 at 12:21
Arrow
5,07621445
5,07621445
add a comment |
add a comment |
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