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Prove that $[0,\infty)$ is not a manifold.

Using diffeomorphisms and the implicit function theorem perhaps.

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    To me manifolds seem a part of algebraic topology and differential geometry. But I'm wondering whether or not to leave the general-topology tag.2012-11-13
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    Of course $[0,\infty)$ is a manifold with boundary. It's just one of the strange twists of mathematical terminology that a manifold with boundary may not be a manifold …2012-11-13

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A topological manifold is a space that looks locally like $\mathbb R^n$. Does $0$ in $[0, \infty)$ look like a point in $\mathbb R$?

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    But I thought you could take a subset of set [0,∞) so (0,∞) to be locally diffeomorphic to R^p ?2012-11-13
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    @Rebekah: look at the definition more carefully.2012-11-13
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    @Rebekah What Chris said. Does your open set contain zero?2012-11-13
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    I think you need to prove that [$0, a)$ cannot be homeomorphic to $\mathbb{R}$ for every $a > 0$. It seems obvious, but the proof does not seem to be so obvious.2012-11-13
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    Think connectedness. What happens when you remove a point from $\mathbb{R}$? What happens when you remove $0$ from $[0,\infty)$?2012-11-13
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    You also need to prove that $[0,\scriptsize+\normalsize\infty)$ is not homeomorphic to $\mathbb{R}^n$ for $\: n\neq 1 \:$. $\hspace{1 in}$2012-11-13
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    @RickyDemer: The same trick goes, just this time removing a point different from $0$ from $[0,\infty)$.2012-11-13
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    @RickyDemer Don't we usually assume that $n$ does not depend on a point of the manifold?2012-11-14
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    @Makoto: $\:\:$ Maybe, but then one also needs to show that one can't use $\:n\neq 1\:$ at (for example) the point $1$. (That's also easy, but not much easier than the rest of the proof.) $\:\:\:$2012-11-14
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    @RickyDemer I know you know the proof. But I would like to prove it for other readers who don't know it. It suffices to prove that $\mathbb{R}$ is not homeomorphic to $\mathbb{R}^n$ for $n>1$. This is because $\mathbb{R}$ minus one point is not connected, while $\mathbb{R}^n$ minus one point is connected.2012-11-15