compact manifold
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From: achille on 11 Jan 2010 05:07 On Jan 11, 5:41 pm, José Carlos Santos <jcsan...(a)fc.up.pt> wrote: > On 10-01-2010 23:23, Maximilian Rogers wrote: > > > > >>>>> How can I show that a compact n- manifold does not embed in R^n? > > >>>> Does the unit sphere S^2, embed into R^2? > > >>> I know that the sphere doesn't embed in R^2, I wouldn't necessarily > >>> know hoe to prove it, though... > > >>> The invariance of dimension theorem says that R^m=R^n iff m=n, right? > > >> That's one very special case of the invariance of domain theorem. > > >>> I am not sure how to use it to prove what i want..could you give me > >>> more details, please? > > >> First you need to find out what the theorem actually says. I could > >> just tell you, but since you evidently have access to the internet > >> it would probably be more instructive for you to simply look it > >> up. On wikipedia for example. > > > Thank you. I was looking up invariance of dimension; only now I looked > > up invariance of domain. > > Then you should be able now to prove that the image of an embedding of > a n-dimensional manifold M into R^n is an open subset of R^n. > > Best regards, > > Jose Carlos Santos I try to follow the idea but there is one point confusing me. We know that the image of an open neighbourhood from the source compact manifold into R^n is open in the subspace topology of R^n, but how can we know it is open in the topology of R^n itself????
From: José Carlos Santos on 11 Jan 2010 06:32 On 11-01-2010 10:07, achille wrote: >>>>>>> How can I show that a compact n- manifold does not embed in R^n? >> >>>>>> Does the unit sphere S^2, embed into R^2? >> >>>>> I know that the sphere doesn't embed in R^2, I wouldn't necessarily >>>>> know hoe to prove it, though... >> >>>>> The invariance of dimension theorem says that R^m=R^n iff m=n, right? >> >>>> That's one very special case of the invariance of domain theorem. >> >>>>> I am not sure how to use it to prove what i want..could you give me >>>>> more details, please? >> >>>> First you need to find out what the theorem actually says. I could >>>> just tell you, but since you evidently have access to the internet >>>> it would probably be more instructive for you to simply look it >>>> up. On wikipedia for example. >> >>> Thank you. I was looking up invariance of dimension; only now I looked >>> up invariance of domain. >> >> Then you should be able now to prove that the image of an embedding of >> a n-dimensional manifold M into R^n is an open subset of R^n. > > I try to follow the idea but there is one point confusing me. > We know that the image of an open neighbourhood from the source > compact manifold into R^n is open in the subspace topology of R^n, > but how can we know it is open in the topology of R^n itself???? If S is a subspace of R^n and S contains an open subset A (I mean, open in S) such that A is homeomorphic to R^n then, by invariance of domain, the inclusion of A in R^n is an open map and therefore A is an open subset of R^n. Best regards, Jose Carlos Santos
From: achille on 11 Jan 2010 08:02 On Jan 11, 7:32 pm, José Carlos Santos <jcsan...(a)fc.up.pt> wrote: > On 11-01-2010 10:07, achille wrote: > > > > >>>>>>> How can I show that a compact n- manifold does not embed in R^n? > > >>>>>> Does the unit sphere S^2, embed into R^2? > > >>>>> I know that the sphere doesn't embed in R^2, I wouldn't necessarily > >>>>> know hoe to prove it, though... > > >>>>> The invariance of dimension theorem says that R^m=R^n iff m=n, right? > > >>>> That's one very special case of the invariance of domain theorem. > > >>>>> I am not sure how to use it to prove what i want..could you give me > >>>>> more details, please? > > >>>> First you need to find out what the theorem actually says. I could > >>>> just tell you, but since you evidently have access to the internet > >>>> it would probably be more instructive for you to simply look it > >>>> up. On wikipedia for example. > > >>> Thank you. I was looking up invariance of dimension; only now I looked > >>> up invariance of domain. > > >> Then you should be able now to prove that the image of an embedding of > >> a n-dimensional manifold M into R^n is an open subset of R^n. > > > I try to follow the idea but there is one point confusing me. > > We know that the image of an open neighbourhood from the source > > compact manifold into R^n is open in the subspace topology of R^n, > > but how can we know it is open in the topology of R^n itself???? > > If S is a subspace of R^n and S contains an open subset A (I mean, open > in S) such that A is homeomorphic to R^n then, by invariance of domain, > the inclusion of A in R^n is an open map and therefore A is an open > subset of R^n. > > Best regards, > > Jose Carlos Santos Thanks, I get it now. I'm confused between the invariance of dimension and invariance of domain.
From: Maximilian Rogers on 11 Jan 2010 12:27 On Jan 11, 7:02 am, achille <achille_...(a)yahoo.com.hk> wrote: > On Jan 11, 7:32 pm, José Carlos Santos <jcsan...(a)fc.up.pt> wrote: > > > > > On 11-01-2010 10:07, achille wrote: > > > >>>>>>> How can I show that a compact n- manifold does not embed in R^n? > > > >>>>>> Does the unit sphere S^2, embed into R^2? > > > >>>>> I know that the sphere doesn't embed in R^2, I wouldn't necessarily > > >>>>> know hoe to prove it, though... > > > >>>>> The invariance of dimension theorem says that R^m=R^n iff m=n, right? > > > >>>> That's one very special case of the invariance of domain theorem. > > > >>>>> I am not sure how to use it to prove what i want..could you give me > > >>>>> more details, please? > > > >>>> First you need to find out what the theorem actually says. I could > > >>>> just tell you, but since you evidently have access to the internet > > >>>> it would probably be more instructive for you to simply look it > > >>>> up. On wikipedia for example. > > > >>> Thank you. I was looking up invariance of dimension; only now I looked > > >>> up invariance of domain. > > > >> Then you should be able now to prove that the image of an embedding of > > >> a n-dimensional manifold M into R^n is an open subset of R^n. > > > > I try to follow the idea but there is one point confusing me. > > > We know that the image of an open neighbourhood from the source > > > compact manifold into R^n is open in the subspace topology of R^n, > > > but how can we know it is open in the topology of R^n itself???? > > > If S is a subspace of R^n and S contains an open subset A (I mean, open > > in S) such that A is homeomorphic to R^n then, by invariance of domain, > > the inclusion of A in R^n is an open map and therefore A is an open > > subset of R^n. > > > Best regards, > > > Jose Carlos Santos > > Thanks, I get it now. I'm confused between the > invariance of dimension and invariance of domain. Yeah, I was confused about the same. I get the argument up to here. WHat is the contradiction, though? Can it not be open because it is compact or something like that?
From: A N Niel on 11 Jan 2010 12:50 > > Yeah, I was confused about the same. I get the argument up to here. > WHat is the contradiction, though? Can it not be open because it is > compact or something like that? It is both open and closed. Does that mean something?
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