DoMath (토론)님의 2007년 2월 14일 (수) 21:48 판
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Set of examples

정리나 성질의 확장, 일반화

  • Heron optical maxmin - steiner network
  • Integral의 정의/개념,

비슷한 증명 Idea

  • Heron's maxmin : Torricelli-Fermat-Steiner's Point, Fagnano-Schwarts triangle,

그러나 general case 가 특별한 경우와 수학적 본질이 많이 다른 경우가 있다. 어떤 경우는 특이한 경우가 더 먼저 증명되고 일반화된 정리가 증명되고 어떤 경우는 그 반대다.

(반대 경우)

The original phrasing was as follows:

Consider a compact 3-dimensional manifold V without boundary. Is it possible that the fundamental group of V could be trivial, even though V is not homeomorphic to the 3-dimensional sphere?

Poincaré never declared whether he believed this additional condition would characterize the 3-sphere, but nonetheless, the statement that it does is known as the Poincaré conjecture. Here is the standard form of the conjecture:

Every simply connected compact 3-manifold (without boundary) is homeomorphic to a 3-sphere.

The conjecture can be generalized to other dimensions, and was solved much earlier (it has a substantially different flavor)


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