How to understand this example in Do Carmo?











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I'm reading the book $Riemannian$ $Geometry$ written by Do Carmo. Here is an example that I cannot understand the explanation he gave.
enter image description here



enter image description here



enter image description here



I really don't understand what he said about why $alpha$ is not an embedding...
No worry about my knowledge on topology, can anyone help me ''translate'' it to the common language that's easy to understand?






general-topology differential-geometry riemannian-geometry curves







share|cite|improve this question




























    up vote
    5
    down vote

    favorite












    I'm reading the book $Riemannian$ $Geometry$ written by Do Carmo. Here is an example that I cannot understand the explanation he gave.
    enter image description here



    enter image description here



    enter image description here



    I really don't understand what he said about why $alpha$ is not an embedding...
    No worry about my knowledge on topology, can anyone help me ''translate'' it to the common language that's easy to understand?






    general-topology differential-geometry riemannian-geometry curves







    share|cite|improve this question


























      up vote
      5
      down vote

      favorite









      up vote
      5
      down vote

      favorite











      I'm reading the book $Riemannian$ $Geometry$ written by Do Carmo. Here is an example that I cannot understand the explanation he gave.
      enter image description here



      enter image description here



      enter image description here



      I really don't understand what he said about why $alpha$ is not an embedding...
      No worry about my knowledge on topology, can anyone help me ''translate'' it to the common language that's easy to understand?






      general-topology differential-geometry riemannian-geometry curves







      share|cite|improve this question















      I'm reading the book $Riemannian$ $Geometry$ written by Do Carmo. Here is an example that I cannot understand the explanation he gave.
      enter image description here



      enter image description here



      enter image description here



      I really don't understand what he said about why $alpha$ is not an embedding...
      No worry about my knowledge on topology, can anyone help me ''translate'' it to the common language that's easy to understand?






      general-topology differential-geometry riemannian-geometry curves




      general-topology differential-geometry riemannian-geometry curves






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited yesterday









      José Carlos Santos

      137k17110199




      137k17110199










      asked yesterday









      user450201

      648




      648






















          1 Answer
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          The set $C=alphabigl((-3,0)bigr)$ has two topologies:




          • the topology it inherits from the usual topology in $mathbb{R}^2$: a set $Asubset C$ is open if there is an open subset of $A^star$ of $mathbb{R}^2$ such that $A=A^starcap C$.

          • the topology in gets from $(-3,0)$: a set $Asubset C$ is open if there is an open subcet $A^star$ of $(-3,0)$ such that $A=alpha(A^star)$.


          Then, Do Carmo explains why these two topologies are distinct: the second one is locally connected, whereas the first one is no.






          share|cite|improve this answer



















          • 1




            Can I just say that (-3,0) is locally connected while the image of alpha is not, so alpha is not a homeomorphism?
            – user450201
            yesterday










          • That would be correct. On the other hand, that is what Do Carmo is claiming.
            – José Carlos Santos
            yesterday











          Your Answer





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          1 Answer
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          active

          oldest

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          active

          oldest

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          active

          oldest

          votes








          up vote
          8
          down vote



          accepted










          The set $C=alphabigl((-3,0)bigr)$ has two topologies:




          • the topology it inherits from the usual topology in $mathbb{R}^2$: a set $Asubset C$ is open if there is an open subset of $A^star$ of $mathbb{R}^2$ such that $A=A^starcap C$.

          • the topology in gets from $(-3,0)$: a set $Asubset C$ is open if there is an open subcet $A^star$ of $(-3,0)$ such that $A=alpha(A^star)$.


          Then, Do Carmo explains why these two topologies are distinct: the second one is locally connected, whereas the first one is no.






          share|cite|improve this answer



















          • 1




            Can I just say that (-3,0) is locally connected while the image of alpha is not, so alpha is not a homeomorphism?
            – user450201
            yesterday










          • That would be correct. On the other hand, that is what Do Carmo is claiming.
            – José Carlos Santos
            yesterday















          up vote
          8
          down vote



          accepted










          The set $C=alphabigl((-3,0)bigr)$ has two topologies:




          • the topology it inherits from the usual topology in $mathbb{R}^2$: a set $Asubset C$ is open if there is an open subset of $A^star$ of $mathbb{R}^2$ such that $A=A^starcap C$.

          • the topology in gets from $(-3,0)$: a set $Asubset C$ is open if there is an open subcet $A^star$ of $(-3,0)$ such that $A=alpha(A^star)$.


          Then, Do Carmo explains why these two topologies are distinct: the second one is locally connected, whereas the first one is no.






          share|cite|improve this answer



















          • 1




            Can I just say that (-3,0) is locally connected while the image of alpha is not, so alpha is not a homeomorphism?
            – user450201
            yesterday










          • That would be correct. On the other hand, that is what Do Carmo is claiming.
            – José Carlos Santos
            yesterday













          up vote
          8
          down vote



          accepted







          up vote
          8
          down vote



          accepted






          The set $C=alphabigl((-3,0)bigr)$ has two topologies:




          • the topology it inherits from the usual topology in $mathbb{R}^2$: a set $Asubset C$ is open if there is an open subset of $A^star$ of $mathbb{R}^2$ such that $A=A^starcap C$.

          • the topology in gets from $(-3,0)$: a set $Asubset C$ is open if there is an open subcet $A^star$ of $(-3,0)$ such that $A=alpha(A^star)$.


          Then, Do Carmo explains why these two topologies are distinct: the second one is locally connected, whereas the first one is no.






          share|cite|improve this answer














          The set $C=alphabigl((-3,0)bigr)$ has two topologies:




          • the topology it inherits from the usual topology in $mathbb{R}^2$: a set $Asubset C$ is open if there is an open subset of $A^star$ of $mathbb{R}^2$ such that $A=A^starcap C$.

          • the topology in gets from $(-3,0)$: a set $Asubset C$ is open if there is an open subcet $A^star$ of $(-3,0)$ such that $A=alpha(A^star)$.


          Then, Do Carmo explains why these two topologies are distinct: the second one is locally connected, whereas the first one is no.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited yesterday

























          answered yesterday









          José Carlos Santos

          137k17110199




          137k17110199








          • 1




            Can I just say that (-3,0) is locally connected while the image of alpha is not, so alpha is not a homeomorphism?
            – user450201
            yesterday










          • That would be correct. On the other hand, that is what Do Carmo is claiming.
            – José Carlos Santos
            yesterday














          • 1




            Can I just say that (-3,0) is locally connected while the image of alpha is not, so alpha is not a homeomorphism?
            – user450201
            yesterday










          • That would be correct. On the other hand, that is what Do Carmo is claiming.
            – José Carlos Santos
            yesterday








          1




          1




          Can I just say that (-3,0) is locally connected while the image of alpha is not, so alpha is not a homeomorphism?
          – user450201
          yesterday




          Can I just say that (-3,0) is locally connected while the image of alpha is not, so alpha is not a homeomorphism?
          – user450201
          yesterday












          That would be correct. On the other hand, that is what Do Carmo is claiming.
          – José Carlos Santos
          yesterday




          That would be correct. On the other hand, that is what Do Carmo is claiming.
          – José Carlos Santos
          yesterday


















           

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