Surface (topology)
Extrinsically defined surfaces and embeddings

Historically, surfaces were initially defined as subspaces of Euclidean spaces. Often, these surfaces were the locus of 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: 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; on the other hand, the real projective plane, which is compact, non-orientable and without boundary, cannot be embedded into E (see Gramain). Steiner surfaces, including Boy's surface, the Roman surface and the cross-cap, are models of the real projective plane in E, but only the Boy surface is an immersed surface. All these models are singular at points where they intersect themselves.

The Alexander horned sphere is a well-known pathological embedding of the two-sphere into the three-sphere.

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 in the "standard" manner (which looks like a bagel) or in a knotted manner (see figure). The two embedded tori are homeomorphic, but not isotopic: They are topologically equivalent, but their embeddings are not.

The image of a continuous, injective function from R to higher-dimensional R is said to be a parametric surface. Such an image is so-called because the x - and y - directions of the domain R 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 to R whose gradient is nowhere zero, then the locus of 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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