View source for Infinitely-distant elements

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''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|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|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|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|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|Aleksandrov compactification]])  $  {\mathbf C  ^ {n} } tilde $, 
homeomorphic to  $  {\mathbf R  ^ {2n} } tilde $
or to the sphere  $  S  ^ {2n} $, 
is also used.

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