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Define a topological manifold as a space locally homeomorphic to $\mathbb{R}^n$. Find a topological manifold that is not Hausdorff and not locally compact. (Hint:Consider $\mathbb{R}$ with "extra origins.") [1]

I've been trying to construct the real line with an extra origin:
Consider $\mathbb{R}^\prime=\mathbb{R}\cup \{0^\prime\}$. If $U$ is open in $\mathbb{R}$, we define open sets of $\mathbb{R}^\prime$ as $U\cup \{0^\prime\}$ if $0\in U$ and $U$ otherwise. I can show that this is a topology, however how do I show that $\mathbb{R}^\prime$ is locally homeomorphic to $\mathbb{R}$?

Any $x\ne 0^\prime$ has an open neighborhood with the identity map to itself as the homeomorphism. However, what is the homeomorphism $h:U\cup \{0^\prime\}\rightarrow \mathbb{R}$ for open sets that contain $0^\prime$?

PS: locally compact is defined as "each point has a neighborhood whose closure is compact."

[1] Abraham, Ralph, and Jerrold E. Marsden. Foundations of mechanics. No. 364. American Mathematical Soc., 2008.

Ali
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    https://math.stackexchange.com/q/1038320/1107685 – Zoe Allen Jul 12 '24 at 03:41
  • It is certainly true that the real line with an extra origin is a non-Hausdorff manifold. But it is locally compact (at least in the usual sense). Any topological manifold is locally compact. This is a bit strange. – David Gao Jul 12 '24 at 04:32
  • @DavidGao There is an answer here: https://math.stackexchange.com/q/2404301/172289 that discusses the compactness issue you mentioned. However, I don't fully understand that answer yet ! – Ali Jul 12 '24 at 04:58
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    @Ali Hmm, this is a different definition of local compactness. Usually local compactness means every point has a neighborhood basis consisting of compact sets, in which case every topological manifold is locally compact. The answer you linked to, however, defines local compactness as every point having an open neighborhood whose closure is compact. The two definitions are equivalent if the space is Hausdorff, but without Hausdorff condition they are not. As the answer you linked to shows, a non-Hausdorff topological manifold can be not locally compact under the second definition. – David Gao Jul 12 '24 at 05:12

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