View source for 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|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i0509001.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i0509002.png" />) are introduced, the number axis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i0509003.png" /> is completed to the compact extended number axis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i0509004.png" />, which is homeomorphic to a (closed) segment. Another way of compactification consists in imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i0509005.png" /> in the real projective straight line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i0509006.png" />, which is homeomorphic to the circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i0509007.png" /> (cf. [[Projective space|Projective space]]); <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i0509008.png" /> is then completed by a single, unique infinitely-distant point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i0509009.png" />.

2) By the addition of a single, unique infinitely-distant point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i05090010.png" /> the finite complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i05090011.png" /> is completed to the compact extended complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i05090012.png" />, which is homeomorphic to the complex [[Projective straight line|projective straight line]] or the Riemann sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i05090013.png" /> (the unit sphere in the Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i05090014.png" />).

3) By the addition of a single, unique infinitely-distant point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i05090015.png" /> the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i05090016.png" />-dimensional real number space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i05090017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i05090018.png" />, is completed to the compact extended number space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i05090019.png" />, which is homeomorphic to the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i05090020.png" />; this homeomorphism can be demonstrated visually by [[Stereographic projection|stereographic projection]]. Another method of compactification consists in imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i05090021.png" /> in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i05090022.png" />-dimensional real projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i05090023.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i05090024.png" />, these two compactifications are not homeomorphic.

For instance, to parallel straight lines, in the projective plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i05090025.png" /> the same infinitely-distant point corresponds, while different infinitely-distant points correspond to non-parallel straight lines. All infinitely-distant points of the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i05090026.png" /> constitute the infinitely-distant straight line. In a similar way, each plane in the projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i05090027.png" /> is completed by an infinitely-distant straight line. All infinitely-distant points and infinitely-distant straight lines in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i05090028.png" /> constitute the infinitely-distant plane. In general, the infinitely-distant elements in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i05090029.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i05090030.png" /> constitute the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i05090032.png" />-dimensional infinitely-distant hyperplane.

4) A compactification of the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i05090033.png" />-dimensional number space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i05090034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i05090035.png" />, is also possible by imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i05090036.png" /> in the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i05090037.png" />-dimensional projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i05090038.png" />. In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i05090039.png" />, too, all infinitely-distant elements of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i05090040.png" /> constitute the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i05090042.png" />-dimensional infinitely-distant hyperplane. Another method of compactification consists in extending <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i05090043.png" /> to the extended complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i05090044.png" />, which is the topological product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i05090045.png" /> copies of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i05090046.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i05090047.png" />, the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i05090048.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i05090049.png" /> are not homeomorphic. The infinitely-distant points of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i05090050.png" /> are the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i05090051.png" /> in which at least one coordinate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i05090052.png" />. The set of all infinitely-distant points of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i05090053.png" /> is naturally subdivided into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i05090054.png" /> sets

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i05090055.png" /></td> </tr></table>

each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i05090056.png" /> having dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i05090057.png" />. The point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i05090058.png" /> belongs to all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i05090059.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i05090060.png" />. For real functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i05090061.png" />, the one-point compactification (cf. [[Aleksandrov compactification|Aleksandrov compactification]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i05090062.png" />, homeomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i05090063.png" /> or to the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050900/i05090064.png" />, 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>