8.6 Topology 525 Exercises Warm Up Exercises In Exercises 1– 6, fill in the blanks with an appropriate word, phrase, or symbol(s). 1. Because it deals with bending and stretching of geometric figures, topology is sometimes referred to as ________ sheet geometry. Rubber 2. A one-sided, one-edged surface is a(n) ________ strip. Möbius 3. A topological object that resembles a bottle but has only one side is a(n) ________ bottle. Klein 4. If you color a map of the United States, the maximum number of colors needed so that no two states that share a common border have the same color is ________. Four 5. A topological object that can be thought of as a circle twisted out of shape is a(n) ________ curve. Jordan 6. The number of holes that go through an object determines the ________ of the object. Genus Practice the Skills In Exercises 7–10, color the map by using a maximum of four colors so that no two regions with a common border have the same color. 7. 1 2 3 7 6 5 4 Answers will vary. 8. 1 2 3 4 5 10 6 7 8 9 Answers will vary. 12. TX OK KS MSAL LA AR MO GA FL SC NC VA KY TN Answers will vary. 13. YT NT NU BC AB SK MB ON QC Answers will vary. 14. BCA BCS SON CHH SIN DGO NAY ZAC SLP TMP NLE COA Answers will vary. Jordan Curve In Exercises 15–20, determine if the point is inside or outside the Jordan curve. 15. Point A Outside 16. Point B Inside 17. Point C Inside 18. Point D Outside 19. Point E Inside 20. Point F Outside Genus In Exercises 21–32, give the genus of the object. If the object has a genus larger than 5, write “larger than 5.” 21. 1 22. 5 D F E B A C SECTION 8.6 9. 3 4 5 6 7 8 1 2 Answers will vary. 10. 1 4 3 2 7 9 6 8 5 Answers will vary. Using the Four-Color Theorem In Exercises 11–14, maps show certain areas of the United States, Canada, and Mexico. Shade in the states (or provinces) using a maximum of four colors so that no two states (or provinces) with a common border have the same color. 11. WA MT ID OR NV CA AZ NM CO UT WY Answers will vary.
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