8.6 Topology 523 (a) (b) Figure 8.94 Jordan Curves A Jordan curve is a topological object that can be thought of as a circle twisted out of shape; see Fig. 8.95 (a)–(d). Like a circle, a Jordan curve has an inside and an outside. To get from a point on the inside of the curve to a point on the outside of the curve, or vice versa, at least one edge of the curve must be crossed. Consider the Jordan curve in Fig. 8.95(d). In our next example, we will determine whether points A and B are inside or outside the Jordan curve. (a) (c) (d) (b) A B Figure 8.95 Example 1 A Jordan Curve Consider the Jordan curve in Fig. 8.95(d). a) Is point A on the inside or the outside of the Jordan curve? b) Is point B on the inside or the outside of the Jordan curve? Solution a) To determine whether a point is inside or outside a Jordan curve, draw a straight line from the point in question to a point that is clearly outside the Jordan curve. In the following figure, we draw two such lines, one from point A and one from point B. A B Four Color Challenge RECREATIONAL MATH Topological Paper Constructions In addition to the Möbius strip, many other interesting surfaces can be made using paper, scissors, and cellophane tape. Shown above is a surface that shares some of the same characteristics as a Möbius strip, but is not topologically equivalent to a Möbius strip. Notice that, like a Möbius strip, this surface has 1 side and 1 edge. Construct the surface shown above using two strips of paper, scissors, and tape. Attempt to cut the surface “in half” by making a small slit along a dashed line in the middle of the paper surface. Then cut along the dashed line shown in the figure, keeping the scissors the same distance from the edge. What happens? Exercise 43 describes another topological construction. Many Internet websites are devoted to topological constructions. One such website demonstrates a Möbius strip made with a zipper. This process allows you to “cut” the Möbius strip and then put the two parts back together again.
RkJQdWJsaXNoZXIy NjM5ODQ=