:: : In the 20th century, the Jordan curve theorem became a subject of intense study, because it was related to the formal axiomatization of topology.
32.
: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.
33.
Since ? does not separate the surface, there is a smooth Jordan curve ? ( with non-vanishing derivative ) which cuts ? transversely at only one point.
34.
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 ).
35.
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.
36.
In 1920, BronisBaw Knaster and Kazimierz Kuratowski asked whether a nondegenerate homogeneous continuum in the Euclidean plane "'R "'2 must be a Jordan curve.
37.
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.
38.
:: : But the proof of the Jordan curve theorem for " continuous curves " without assuming differentiability, is more subtle, because continuous curves can be complicated.
39.
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.
40.
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.
