Locally Euclidean topological space

From Maths
Jump to: navigation, search
Stub grade: A**
This page is a stub
This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:
This is a notes quality page it is actively being worked upon and should not be taken to be as reliable as a normal page
  • I am currently doing the proofs for equivalent definitions Alec (talk) 16:55, 19 February 2017 (UTC)
Caveat:Need to do locally euclidean of dimension [ilmath]n[/ilmath]!


Let [ilmath](X,\mathcal{ J })[/ilmath] be a topological space, we say it is locally Euclidean if:

  • [ilmath]\forall p\in X\exists n\in\mathbb{N}_0\exists U\in\mathcal{O}(p;X)\underbrace{\exists V\in\mathcal{O}(\mathbb{R}^n)\exists\varphi\in\mathcal{F}(U,V)[U\cong_\varphi V]}_{\exists\text{homeomorphism }\varphi:U\rightarrow\text{some open subset of }\mathbb{R}^n } [/ilmath]
    • In words: for all points, [ilmath]p\in X[/ilmath], there is an [ilmath]n[/ilmath] such that there is an open neighbourhood of [ilmath]p[/ilmath] which is homeomorphic to some open set in [ilmath]\mathbb{R}^n[/ilmath] for the given {{M|n}].

Equivalent definitions

Some open ball at the origin


  • [ilmath]\Big(\forall p\in X\exists n\in\mathbb{N}_0\exists U\in\mathcal{O}(p;X)\exists V\in\mathcal{O}(\mathbb{R}^n)\exists\varphi\in\mathcal{F}(U,V)[U\cong_\varphi V]\Big)[/ilmath] [ilmath]\iff[/ilmath] [ilmath]\Big(\forall p\in X\exists n\in\mathbb{N}_0\exists U\in\mathcal{O}(p;X)\exists\epsilon\in\mathbb{R}_{>0}\exists\varphi\in\mathcal{F}(U,B_\epsilon(0;\mathbb{R}^n)[U\cong_\varphi B_\epsilon(0;\mathbb{R}^n)]\Big)[/ilmath]



  • Let [ilmath]p\in X[/ilmath] be given
    • Choose [ilmath]n:\eq n'[/ilmath] where [ilmath]n'[/ilmath] is the [ilmath]n\in\mathbb{N}_0[/ilmath] posited to exist by the LHS of the [ilmath]\iff[/ilmath]
      • We now obtain:
        • [ilmath]U'\in\mathcal{O}(p;X)[/ilmath] posited to exist by the LHS,
          • [ilmath]V'\in\mathcal{O}(\mathbb{R}^n)[/ilmath] posited to exist by the LHS, such that:
            • [ilmath]\varphi':U'\rightarrow V'[/ilmath] posited to exist by the LHS is a homeomorphism.
      • Recall that "an open set in a metric space contains an open ball about all of its points", this means:
        • [ilmath]\exists\delta\in\mathbb{R}_{>0}[B_\delta(\varphi'(p);\mathbb{R}^n)\subseteq V'][/ilmath]
      • As open balls are open sets and "the pre-image of an open set under a homeomorphism is open" we see:
        • [ilmath]\varphi'^{-1}\big(B_\delta(\varphi'(p);\mathbb{R}^n)\big)[/ilmath] is open in [ilmath]X[/ilmath]
      • We must now show that [ilmath]p\in\varphi'^{-1}\big(B_\delta(\varphi'(p);\mathbb{R}^n)\big)[/ilmath] (so we can say [ilmath]\varphi'^{-1}\big(B_\delta(\varphi'(p);\mathbb{R}^n)\big)\in\mathcal{O}(p,X)[/ilmath] shortly)
        • Note that [ilmath]f(p)\in B_\delta(\varphi'(p);\mathbb{R}^n)[/ilmath] as [ilmath]d(\varphi'(p),\varphi'(p)):\eq 0[/ilmath] regardless of the metric used
          • As [ilmath]0<\delta[/ilmath] we see [ilmath]\varphi'(p)\in B_\delta(\varphi'(p);\mathbb{R}^n)[/ilmath]
            • Thus [ilmath]p\in\varphi'^{-1}\big(B_\delta(\varphi'(p);\mathbb{R}^n)\big)[/ilmath] - by the definition of pre-image
      • Choose: [ilmath]U:\eq \varphi'^{-1}\big(B_\delta(\varphi'(p);\mathbb{R}^n)\big)[/ilmath], by the discussion above we see [ilmath]U\in\mathcal{O}(p,X)[/ilmath]
        • Choose: [ilmath]\epsilon\eq\delta[/ilmath], we found [ilmath]\delta[/ilmath] above, recall [ilmath]\delta\in\mathbb{R}_{>0} [/ilmath], so obviously [ilmath]\epsilon\in\mathbb{R}_{>0} [/ilmath]
Grade: A
This page requires some work to be carried out
Some aspect of this page is incomplete and work is required to finish it
The message provided is:
The bulk of the proof is done here, the "not trivial part" is that [ilmath]\varphi'[/ilmath] restricts to a homeomorphism onto [ilmath]B_\delta(\varphi'(p);\mathbb{R}^n)[/ilmath], then compose that with a translation, we use translations are homeomorphisms and we're basically done. Alec (talk) 16:55, 19 February 2017 (UTC)