Word.ws English Dictionary

complex projective line

complex projective line noun: a complex line (especially, the set of complex numbers regarded as such) endowed with a point at infinity (thus becoming a projective line); (more formally) the set of equivalence classes of ordered pairs (α, β) of complex numbers, not both zero, with respect to the equivalence relation "(α, β) ≡ (λα, λβ) for all nonzero complex λ".

Noun

complex projective line (plural complex projective lines)

  1. (analytic, projective geometry) A complex line (especially, the set of complex numbers regarded as such) endowed with a point at infinity (thus becoming a projective line); (more formally) the set of equivalence classes of ordered pairs (α, β) of complex numbers, not both zero, with respect to the equivalence relation "(α, β) ≡ (λα, λβ) for all nonzero complex λ".
    • By way of an example, we consider in detail the complex projective line #92;C#92;mathbb#123;P#125;¹.[…]Thus via the function w#95;0 the complex projective line #92;C#92;mathbb#123;P#125;¹ becomes identified with the "extended complex plane" (i.e. the ordinary complex plane with an additional "point...
    • 2..2.1 Theorem The complex projective line #92;C#92;mathbb#123;P#125;¹ is diffeomorphic to the 2-dimensional sphere S².1985, B. A. Dubrovin, A. T. Fomenko, S. P. Novikov, translated by R. G. Burns, Modern Geometry — Methods and Applications: Part II, Springer, page 18
    Synonyms:
    extended complex numbers, Riemann sphere

Examples of complex projective line in a sentence

  • Finally, C is interested in the geometry of the complex projective line P¹(#92;C).2009, Jean-Pierre Marquis, From a Geometrical Point of View: A Study of the History and Philosophy of Category Theory, Springer, page 14:
  • Hence, C(#92;Gamma) is an uncountable union of complex projective lines.2013, Angel Cano, Juan Pablo Navarrete, José Seade, Complex Kleinian Groups, Springer (Birkhäuser), page 142:

Synonyms of complex projective line

Forms of complex projective line

pluralcomplex projective lines

Questions about complex projective line

What does complex projective line mean?
Complex projective line means a complex line (especially, the set of complex numbers regarded as such) endowed with a point at infinity (thus becoming a projective line); (more formally) the set of equivalence classes of ordered pairs (α, β) of complex numbers, not both zero, with respect to the equivalence relation "(α, β) ≡ (λα, λβ) for all nonzero complex λ".
How do you use complex projective line in a sentence?
Finally, C is interested in the geometry of the complex projective line P¹(#92;C).
What part of speech is complex projective line?
Complex projective line is a noun.
What is the plural of complex projective line?
The plural of complex projective line is complex projective lines.

Words near complex projective line in the dictionary

  1. complex number
  2. complex numbers
  3. complex plane
  4. complex planes
  5. complex post-traumatic stress disorder
  6. complex projective line
  7. complex projective lines
  8. complex question
  9. complex questions
  10. complex regional pain syndrome
  11. complex regional pain syndromes

Source and license

Adapted from Wiktionary: complex projective line (page history and authors). Text available under the Creative Commons Attribution-ShareAlike 4.0 license. Word.ws reformatted the entry and added links. Wiktionary data of 2026-09-02.

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