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I just started reading a textbook, and it keeps saying that an $n$-dimensional manifold is a topological space with the same local properties as Euclidean $n$-space.

I don't really understand what is meant by "local properties." What what would be an example for, say, a $2$-dimensional manifold that is not a Euclidean $2$-space like $\mathbb{R}^2$ or something, but still has the "local properties" of the Euclidean space?

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    $\begingroup$ "Local properties" is too vague to give an answer to. What it means is that for any point $p$ there is an open set $p \in U \subset M$ with $U$ homeomorphic to $\Bbb R^n$. The 2-sphere satisfied this, but is not homeomorphic to $\Bbb R^2$. $\endgroup$
    – user98602
    Aug 25 '15 at 22:37
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    $\begingroup$ How about $S^1 \times S^1$ inside $\mathbb{C}^2$. This is a torus. $\endgroup$ Aug 25 '15 at 22:37
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The surface of a sphere, $S^2$, is a manifold that locally looks flat (if you get down to level earth, wouldn't you say the Earth looks flat?), yet obviously is not globally flat.

What it means, heuristically, is that you can approximate the neighborhood of any point with a flat plane (or, generally with the appropriate $\mathbb{R}^n$), and "do math" on the manifold essentially by doing math in that plane, provided you pay proper attention to how that tangent plane changes as you move around the manifold.

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