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In the notes above, I am not quite sure what the up-side-down $\Pi$ is, nor the notation of $/ \sim$.

$\amalg$ means disjoint union as Peter Tamaroff advised.

As for the notation of$/ \sim$, I guess it means take away subsets in the manifold that are mapped for multiple times? Therefore, each element in the manifold is covered by a coordinate chart exactly once, hence is disjoint?

What does it mean by "One just need to check the resulting space is Hausdorf" at the end?

The reason it mentions Hausdorff at the end of the remark is because the definition of manifold:

A second-countable Hausdorff space $\mathcal{X}$ is an $n$-dimensional (abstract) manifold if for every $p \in \mathcal{X}$ there exists a neighborhood $\mathcal{U}$ and a homeomorphism $\varphi: \mathcal{U} \rightarrow \mathcal{U}^\prime$ to an open set $\mathcal{U}^\prime \rightarrow \mathbb{R}^n$.

WishingFish
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    Notation $\coprod$ stands for "disjoint union", as the text says. An alternative notation is $\bigsqcup$. – Pedro Jun 24 '13 at 02:30

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The notation "$\coprod$" refers to disjoint union. The notation "$/\sim$" refers to taking the quotient by the equivalence relation $\sim$. That is, you identify those points which are related by $\sim$.

In short, the definition is "take the disjoint sets $\mathcal U'_\alpha$ and for each point $p\in \mathcal U'_\alpha$, glue $p$ to $(\varphi_\beta\circ\varphi_\alpha^{-1})(p)$."

bradhd
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