- #1
NihilTico
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1. Homework Statement
I'm taking a swing at Spivak's Differential Geometry, and a question that Spivak asks his reader to show is that if ##x\in M## for ##M## a manifold and there is a neighborhood (Note that Spivak requires neighborhoods to be sets which contain an open set containing the point of interest) of ##x## such that this neighborhood is homeomorphic to ##\mathbb{R}^n##, then this ##n## is fixed for any other such neighborhood of ##x##.
If I could prove this, then I could prove his other assertion that if ##U\subseteq M## is a neighborhood of some ##x\in M##, and is homeomorphic to ##\mathbb{R}^n## (for any ##n##), then ##U## must be open; via the Invariance of Domain, the proof follows almost immediately from what I'm proving here.
2. The attempt at a solution
I'm a little far removed from topology so I'm not entirely confident in my proof. In particular, when I tried to find some sort of verification online, I found this which doesn't look anything like my argument.
I more or less mimic my proof that ##\mathbb{R}^n## is not homeomorphic to ##\mathbb{R}^m## if ##n\ne m## by the invariance of domain in my proof:Spivak has shown that one can choose, for any neighborhood ##U## of ##x\in M##, a subset ##V\subseteq U## such that ##V## is open with respect to the topology of ##M## and homeomorphic to the same ##\mathbb{R}^n## as ##U##. This is straight-forward.
So if I let ##x\in M##, and ##N_1##, ##N_2## be two neighborhoods of ##x## homeomorphic, respectively, to ##\mathbb{R}^n## and ##\mathbb{R}^m##. Let's denote by ##f\colon N_1\to\mathbb{R}^n## and ##g\colon N_{2}\to\mathbb{R}^m## the homeomorphisms. WLOG we can assume ##n<m##, attempting to obtain a contradiction.
I can widdle ##N_1## and ##N_2## down into subsets ##V_1## and ##V_2## open in ##M## as I noted above. Let ##V=V_1\cap V_2##. Then, of course, ##V\subseteq N_{1}## and ##V\subseteq N_2##; moreover ##V## is still open in ##M##. Since ##g## and ##f## are homeomorphisms, the image of ##V## is open in ##\mathbb{R}^n## under ##f## and the image of ##V## under ##g## is open in ##\mathbb{R}^m##. Denote by ##\pi## the embedding of ##\mathbb{R}^n## into ##\mathbb{R}^m##. Then the map ##h:=\pi\circ f\circ g^{-1}## passes ##g(V)## to ##f(V)\subseteq\mathbb{R}^n## viewed as a subset of ##\mathbb{R}^m##. But in the metric topology of ##\mathbb{R}^m## (where, again, ##n<m##), any subset of ##\mathbb{R}^n## is closed. So this contradicts the invariance of domain, since ##h## defines a continuous injection of an open subset of ##\mathbb{R}^m## into ##\mathbb{R}^m##.
Does this look alright? Any pointers?
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