Infinitely-distant elements

infinitely-remote elements, improper elements, ideal elements

Elements (points, straight lines, planes, etc.), generated by extending a given affine space to a compact space. Infinitely-distant elements are one of the forms in which the "actual" infinity is manifested in various mathematical theories. The continuous connection of the finite and the infinite is manifested by the fact that infinitely-distant elements are meaningful only in as far as they are considered in the context of some concrete compactification of a given "finite" space. The types of infinitely-distant elements resulting from the most frequent methods of compactification of finite-dimensional Euclidean spaces are described below.

1) If infinitely-distant elements (points $ - \infty $ and $ + \infty $) are introduced, the number axis $ \mathbf R $ is completed to the compact extended number axis $ \overline{\mathbf R}\; $, which is homeomorphic to a (closed) segment. Another way of compactification consists in imbedding $ \mathbf R $ in the real projective straight line $ P _ {1} ( \mathbf R ) = \widetilde{\mathbf R} $, which is homeomorphic to the circle $ S ^ {1} $( cf. Projective space); $ \mathbf R $ is then completed by a single, unique infinitely-distant point $ \infty $.

2) By the addition of a single, unique infinitely-distant point $ \infty $ the finite complex plane $ \mathbf C $ is completed to the compact extended complex plane $ \overline{\mathbf C}\; $, which is homeomorphic to the complex projective straight line or the Riemann sphere $ S ^ {2} $( the unit sphere in the Euclidean space $ \mathbf R ^ {3} $).

3) By the addition of a single, unique infinitely-distant point $ \infty $ the $ n $- dimensional real number space $ \mathbf R ^ {n} $, $ n \geq 1 $, is completed to the compact extended number space $ \widetilde{\mathbf R} {} ^ {n} $, which is homeomorphic to the sphere $ S ^ {n} $; this homeomorphism can be demonstrated visually by stereographic projection. Another method of compactification consists in imbedding $ \mathbf R ^ {n} $ in the $ n $- dimensional real projective space $ P _ {n} ( \mathbf R ) $. If $ n > 1 $, these two compactifications are not homeomorphic.

For instance, to parallel straight lines, in the projective plane $ P _ {2} ( \mathbf R ) $ the same infinitely-distant point corresponds, while different infinitely-distant points correspond to non-parallel straight lines. All infinitely-distant points of the plane $ P _ {2} ( \mathbf R ) $ constitute the infinitely-distant straight line. In a similar way, each plane in the projective space $ P _ {3} ( \mathbf R ) $ is completed by an infinitely-distant straight line. All infinitely-distant points and infinitely-distant straight lines in $ P _ {3} ( \mathbf R ) $ constitute the infinitely-distant plane. In general, the infinitely-distant elements in $ P _ {n} ( \mathbf R ) $ of dimension $ \leq ( n - 2) $ constitute the $ ( n - 1) $- dimensional infinitely-distant hyperplane.

4) A compactification of the complex $ n $- dimensional number space $ \mathbf C ^ {n} $, $ n \geq 1 $, is also possible by imbedding $ \mathbf C ^ {n} $ in the complex $ n $- dimensional projective space $ P _ {n} ( \mathbf C ) $. In $ P _ {n} ( \mathbf C ) $, too, all infinitely-distant elements of dimension $ \leq ( n - 2) $ constitute the complex $ ( n - 1) $- dimensional infinitely-distant hyperplane. Another method of compactification consists in extending $ \mathbf C ^ {n} $ to the extended complex space $ \overline{ {\mathbf C ^ {n} }}\; $, which is the topological product of $ n $ copies of $ \overline{\mathbf C}\; $. If $ n > 1 $, the spaces $ P _ {n} ( \mathbf C ) $ and $ \overline{ {\mathbf C ^ {n} }}\; $ are not homeomorphic. The infinitely-distant points of the space $ \overline{ {\mathbf C ^ {n} }}\; $ are the points $ z = ( z _ {1} \dots z _ {n} ) $ in which at least one coordinate $ z _ \nu = \infty $. The set of all infinitely-distant points of the space $ \overline{ {\mathbf C ^ {n} }}\; $ is naturally subdivided into $ n $ sets

$$ M _ \nu = \ \{ {z \in \overline{ {\mathbf C ^ {n} }}\; } : { z _ {r} = \infty , z _ {k} \in \overline{ {\mathbf C }}\; , k \neq r } \} , $$

each $ M _ \nu $ having dimension $ n - 1 $. The point $ ( \infty \dots \infty ) $ belongs to all $ M _ \nu $, $ \nu = 1 \dots n $. For real functions on $ \mathbf C ^ {n} $, the one-point compactification (cf. Aleksandrov compactification) $ {\mathbf C ^ {n} } tilde $, homeomorphic to $ {\mathbf R ^ {2n} } tilde $ or to the sphere $ S ^ {2n} $, is also used.

References

[1]	N. Bourbaki, "Elements of mathematics. General topology" , Springer (1988) (Translated from French)
[2]	N.V. Efimov, "Höhere Geometrie" , Deutsch. Verlag Wissenschaft. (1960) (Translated from Russian)
[3]	R. Hartshorne, "Foundations of projective geometry" , Benjamin (1967)
[4]	B.A. Fuks, "Introduction to the theory of analytic functions of several complex variables" , Amer. Math. Soc. (1965) (Translated from Russian)
[5]	B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1976) (In Russian)