Surface question (topology)

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This is actually a topology question, but I wasn't sure where to ask it. It's about surfaces, ie, 2D manifolds. I know that the defining property of a surface is that each point has a neighborhood homeomorphic to R2, but I was wondering if this neighborhood is always open in the surface. It seems pretty simple, but I can't seem to find a way to prove it. If it isn't true in general, what conditions would make it true?
 

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