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Definition of surface from Pressley's "Elementary differential geometry":

I do not find it evident that this definition reproduces the intuitive notion of a surface. For instance, it is not obvious that for S a "solid" (such as a full sphere: {[itex](x,y,z)\in \mathbb{R}^3:x^2+y^2+z^2\leq R^2[/itex]}), we can't find a collection of homeomorphisms that cover S.

Is there a result somewhere in mathematics that says something like that?

*A subset S of [itex]\mathbb{R}^3[/itex] is a**surface*if, for every point P in S, there is an open set U in [itex]\mathbb{R}^2[/itex] and an open set W in [itex]\mathbb{R}^3[/itex] containnign P such that [itex]S\cap W[/itex] is homeomorphic to U.I do not find it evident that this definition reproduces the intuitive notion of a surface. For instance, it is not obvious that for S a "solid" (such as a full sphere: {[itex](x,y,z)\in \mathbb{R}^3:x^2+y^2+z^2\leq R^2[/itex]}), we can't find a collection of homeomorphisms that cover S.

Is there a result somewhere in mathematics that says something like that?

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