Editing Surface (topology) (section)

==Extrinsically defined surfaces and embeddings==
[[Image:Sphere wireframe.svg|left|thumb|250px|A sphere can be defined parametrically (by ''x'' = ''r'' sin ''θ'' cos ''φ'',
''y'' = ''r'' sin ''θ'' sin ''φ'', ''z'' = ''r'' cos ''θ'') or implicitly (by {{nowrap|''x''<sup>2</sup> + ''y''<sup>2</sup> + ''z''<sup>2</sup> &minus; ''r''<sup>2</sup> {{=}} 0}}.)]]

Historically, surfaces were initially defined as subspaces of Euclidean spaces. Often, these surfaces were the [[locus (mathematics)|locus]] of [[root of a function|zeros]] of certain functions, usually polynomial functions. Such a definition considered the surface as part of a larger (Euclidean) space, and as such was termed ''extrinsic''.

In the previous section, a surface is defined as a topological space with certain properties, namely Hausdorff and locally Euclidean. This topological space is not considered a subspace of another space. In this sense, the definition given above, which is the definition that mathematicians use at present, is ''intrinsic''.

A surface defined as intrinsic is not required to satisfy the added constraint of being a subspace of Euclidean space.  It may seem possible for some surfaces defined intrinsically to not be surfaces in the extrinsic sense. However, the [[Whitney embedding theorem]] asserts every surface can in fact be embedded homeomorphically into Euclidean space, in fact into '''E'''<sup>4</sup>: The extrinsic and intrinsic approaches turn out to be equivalent.

In fact, any compact surface that is either orientable or has a boundary can be embedded in '''E'''<sup>3</sup>; on the other hand, the real projective plane, which is compact, non-orientable and without boundary, cannot be embedded into '''E'''<sup>3</sup> (see Gramain). [[Steiner surface]]s, including [[Boy's surface]], the [[Roman surface]] and the [[cross-cap]], are models of the real projective plane in '''E'''<sup>3</sup>, but only the Boy surface is an [[immersion (mathematics)|immersed surface]]. All these models are singular at points where they intersect themselves.

The [[Alexander horned sphere]] is a well-known [[pathological (mathematics)|pathological]] embedding of the two-sphere into the three-sphere.

[[Image:KnottedTorus.svg|right|thumb|A knotted torus.]]
The chosen embedding (if any) of a surface into another space is regarded as extrinsic information; it is not essential to the surface itself. For example, a torus can be embedded into '''E'''<sup>3</sup> in the "standard" manner (which looks like a [[bagel]]) or in a [[knot (mathematics)|knotted]] manner (see figure). The two embedded tori are homeomorphic, but not [[Homotopy#Isotopy|isotopic]]: They are topologically equivalent, but their embeddings are not.

The [[image (mathematics)|image]] of a continuous, [[injection (mathematics)|injective]] function from '''R'''<sup>2</sup> to higher-dimensional '''R'''<sup>n</sup> is said to be a [[parametric surface]]. Such an image is so-called because the ''x''- and ''y''- directions of the domain '''R'''<sup>2</sup> are 2 variables that parametrize the image. A parametric surface need not be a topological surface. A [[surface of revolution]] can be viewed as a special kind of parametric surface.

If ''f'' is a smooth function from '''R'''<sup>3</sup> to '''R''' whose [[gradient]] is nowhere zero, then the [[locus (mathematics)|locus]] of [[root of a function|zeros]] of ''f'' does define a surface, known as an ''[[implicit surface]]''. If the condition of non-vanishing gradient is dropped, then the zero locus may develop singularities.
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