Curves and Surfaces: Topology
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.
7.1 Curves in the Euclidean Topology
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.
7.1.1 Jordan curves
Let φ be a parameterized continuous path φ :[a, b] → R2 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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