CHAPTER 7

On the basis of the concepts introduced in Chapter 6, this chapter discusses curves and surfaces in topologic spaces, with emphasis on the digital case.

The definition of a curve has an interesting history in mathematics. Jordan curves are defined by parameterization; see the “parameterized paths ” in Section 6.3.3. Urysohn-Menger curves are defined using a topologic approach. The two definitions are equivalent, and they both define separations of the plane.

Let φ be a parameterized continuous path φ :[*a, b*] → R^{2} such that *a* ≠ *b*, φ (*a*) = φ(*b*), and let φ(S) ≠φ(*i*) for all *s,t* (*a* ≤ *s*<*t*<*b*). The following set was defined by C. Jordan in 1893 to be ...

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