Slice Chart Condition Proof - Topological Embedding











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Below is some background information from Lee's Introduction To Smooth Manifolds about slice charts of embedded submanifolds. My question is at the bottom.




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What I don't understand in the proof is the part highlighted in red. I don't understand how we can conclude that $S$ is a topological embedding from what we proved so far.






general-topology differential-geometry manifolds differential-topology smooth-manifolds







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    up vote
    0
    down vote

    favorite












    Below is some background information from Lee's Introduction To Smooth Manifolds about slice charts of embedded submanifolds. My question is at the bottom.




    enter image description here



    enter image description here



    enter image description here



    enter image description here




    What I don't understand in the proof is the part highlighted in red. I don't understand how we can conclude that $S$ is a topological embedding from what we proved so far.






    general-topology differential-geometry manifolds differential-topology smooth-manifolds







    share|cite|improve this question


























      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      Below is some background information from Lee's Introduction To Smooth Manifolds about slice charts of embedded submanifolds. My question is at the bottom.




      enter image description here



      enter image description here



      enter image description here



      enter image description here




      What I don't understand in the proof is the part highlighted in red. I don't understand how we can conclude that $S$ is a topological embedding from what we proved so far.






      general-topology differential-geometry manifolds differential-topology smooth-manifolds







      share|cite|improve this question















      Below is some background information from Lee's Introduction To Smooth Manifolds about slice charts of embedded submanifolds. My question is at the bottom.




      enter image description here



      enter image description here



      enter image description here



      enter image description here




      What I don't understand in the proof is the part highlighted in red. I don't understand how we can conclude that $S$ is a topological embedding from what we proved so far.






      general-topology differential-geometry manifolds differential-topology smooth-manifolds




      general-topology differential-geometry manifolds differential-topology smooth-manifolds






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




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      edited yesterday

























      asked 2 days ago









      Frederic Chopin

      31218




      31218






















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          Up to the point you have highlighted, we have shown that for any point $p$ in $S$, we can define a chart $(psi,V)$, where $V$ is open in $S$ and contains $p$, with $psicolon Vto widehat Vsubset Bbb R^k$ a homeomorphism. The inclusion map $iotacolon Shookrightarrow M$ is a topological embedding because $S$ is given the subspace topology, hence $iota$ is a homeomorphism onto its image.






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            active

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            up vote
            1
            down vote



            accepted










            Up to the point you have highlighted, we have shown that for any point $p$ in $S$, we can define a chart $(psi,V)$, where $V$ is open in $S$ and contains $p$, with $psicolon Vto widehat Vsubset Bbb R^k$ a homeomorphism. The inclusion map $iotacolon Shookrightarrow M$ is a topological embedding because $S$ is given the subspace topology, hence $iota$ is a homeomorphism onto its image.






            share|cite|improve this answer

























              up vote
              1
              down vote



              accepted










              Up to the point you have highlighted, we have shown that for any point $p$ in $S$, we can define a chart $(psi,V)$, where $V$ is open in $S$ and contains $p$, with $psicolon Vto widehat Vsubset Bbb R^k$ a homeomorphism. The inclusion map $iotacolon Shookrightarrow M$ is a topological embedding because $S$ is given the subspace topology, hence $iota$ is a homeomorphism onto its image.






              share|cite|improve this answer























                up vote
                1
                down vote



                accepted







                up vote
                1
                down vote



                accepted






                Up to the point you have highlighted, we have shown that for any point $p$ in $S$, we can define a chart $(psi,V)$, where $V$ is open in $S$ and contains $p$, with $psicolon Vto widehat Vsubset Bbb R^k$ a homeomorphism. The inclusion map $iotacolon Shookrightarrow M$ is a topological embedding because $S$ is given the subspace topology, hence $iota$ is a homeomorphism onto its image.






                share|cite|improve this answer












                Up to the point you have highlighted, we have shown that for any point $p$ in $S$, we can define a chart $(psi,V)$, where $V$ is open in $S$ and contains $p$, with $psicolon Vto widehat Vsubset Bbb R^k$ a homeomorphism. The inclusion map $iotacolon Shookrightarrow M$ is a topological embedding because $S$ is given the subspace topology, hence $iota$ is a homeomorphism onto its image.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered yesterday









                AOrtiz

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