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I'm reading the differential geometry written by DoCarmo and having trouble when understanding the definition of regular surface. What troubles me is that I could not see why the definition would rule out those case when the surface has boundaries.

For example, the set $\lbrace z=0$ and $x^2+y^2 \leq 1\rbrace$. It is said that a regular surface should be locally homeomorphic to an open set in $\mathbb{R}^2$. While since any boundary point is an interior point in the surface itself, I couldn't see why a boundary point cannot exists in a regular surface. Could anyone clear this concept for me? I really appreciate it!

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Think about what a neighborhood of a boundary point in the surface looks like - it should look like a half disk with the boundary diameter. This is the main difference with the non-boundary point, whose small neighborhood in the surface looks exactly like a full disk without boundary. – user27126 Oct 7 '13 at 6:05

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