A topological $n$-manifold $M$ is locally Euclidean of dimension $n$ (each point of $M$ has a neighborhood that is homeomorphic to an open subset of $\mathbb{R}^n)$.

But what does locally Euclidean of dimension 0 mean?

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    $\begingroup$ Discrete${}{}$? $\endgroup$ – Angina Seng Jun 3 '20 at 13:58
  • $\begingroup$ @AnginaSeng What does that mean? $\endgroup$ – Filippo Jun 3 '20 at 14:04
  • $\begingroup$ It means that "locally Euclidean of dimension zero" is equivalent to "discrete". $\endgroup$ – Lee Mosher Jun 3 '20 at 14:45
  • $\begingroup$ @LeeMosher A topological space $M$ is discrete if (and only if) every subset of $M$ is open, right? $\endgroup$ – Filippo Jun 3 '20 at 15:18
  • $\begingroup$ @Filippo right, all subsets are open. However manifolds are requested to be second countable also, hence uncountable discrete spaces are not allowed. $\endgroup$ – InsideOut Jun 3 '20 at 15:28

$\Bbb R^0$ is just a point, isn't it? Hence locally Euclidean of dimension zero means that locally homeomorphic to $\Bbb R^0=\{pt\}$. A $0-$topological manifold is then a countable set endowed with the discrete topology.

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    $\begingroup$ Only countable family if you have having a countable base as a part of the definition of a manifold. The OP might well not assume that. $\endgroup$ – Henno Brandsma Jun 3 '20 at 15:31
  • $\begingroup$ Maybe we are using different definitions of manifolds. I have always assumed manifolds to be second-countable and then the existence of a countable basis. However, if we drop the second-countability condition also uncountable union of points endowed with the discrete topology are allowed. $\endgroup$ – InsideOut Jun 3 '20 at 15:35
  • $\begingroup$ I am currently reading John Lee's book about smooth manifolds and i just found the equation/definition $\mathbb{R}^0=\{0\}$ on page 25. Since $\{\{0\},\emptyset\}$ is the only existing topology on $\{0\}$, we can even make sense of the definition in my question for the case $n=0$ and we can prove what InsideOut said. $\endgroup$ – Filippo Jun 3 '20 at 19:52

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