jordan curve in a sentence

Examples

1. :: : In the 20th century, the Jordan curve theorem became a subject of intense study, because it was related to the formal axiomatization of topology.
2. :When you talk about the topology of space you may get into some difficulties proving things that seem intuitively obvious; see Jordan Curve Theorem as an example.
3. Since ? does not separate the surface, there is a smooth Jordan curve ? ( with non-vanishing derivative ) which cuts ? transversely at only one point.
4. In particular, it is impossible to dissect a circle and make a square using pieces that could be cut with scissors ( that is, having Jordan curve boundary ).
5. Applications of the fundamental groupoid on a set of base points to the Jordan curve theorem, covering spaces, and orbit spaces are given in Ronald Brown's book.
6. In 1920, BronisB
   aw Knaster and Kazimierz Kuratowski asked whether a nondegenerate homogeneous continuum in the Euclidean plane "'R "'2 must be a Jordan curve.
7. In fact suppose a region on the Riemann sphere is given by the exterior of " n " disjoint Jordan curves and that  " is an exterior point.
8. :: : But the proof of the Jordan curve theorem for " continuous curves " without assuming differentiability, is more subtle, because continuous curves can be complicated.
9. Now combining C1 with C satisfies the requirements of the Jordan Curve Theorem and generates two connected components, E1 and E2, with C1 + C as the boundary between them.
10. Due to the importance of the Jordan curve theorem in low-dimensional topology and complex analysis, it received much attention from prominent mathematicians of the first half of the 20th century.