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CHAPTER 7

# 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 ab, φ (a) = φ(b), and let φ(S) ≠φ(i) for all s,t (as<t<b). The following set was defined by C. Jordan in 1893 to be ...

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