Exercises: Chapter 4, Section 3

  1. Let ch4c1.png be the set of all singular ch4c2.png -cubes, and ch4c3.png the integers. An ch4c4.png -chain is a function ch4c5.png such that ch4c6.png for all but finitely many ch4c7.png . Define ch4c8.png and ch4c9.png by ch4c10.png and ch4c11.png . Show that ch4c12.png and ch4c13.png are ch4c14.png -chains if ch4c15.png and ch4c16.png are. If ch4c17.png , let ch4c18.png also denote the function ch4c19.png such that ch4c20.png and ch4c21.png for ch4c22.png . Show that every ch4c23.png -chain ch4c24.png can be written ch4c25.png for some integers ch4c26.png and singular ch4c27.png -cubes ch4c28.png .

    Since ch4c29.png and ch4c30.png , the functions ch4c31.png and ch4c32.png are ch4c33.png -chains if ch4c34.png and ch4c35.png are.

    The second assertion is obvious since ch4c36.png .

  2. For ch4c37.png and ch4c38.png an integer, define the singular 1-cube ch4c39.png by ch4c40.png . Show that there is a singular 2-cube ch4c41.png such that ch4c42.png .

    Define ch4c43.png by ch4c44.png where ch4c45.png and ch4c46.png are positive real numbers. The boundary of ch4c47.png is easily seen to be ch4c48.png .

  3. If ch4c49.png is a singular 1-cube in ch4c50.png with ch4c51.png , show that there is an integer ch4c52.png such that ch4c53.png for some 2-chain ch4c54.png .

    Given ch4c55.png , let ch4c56.png where ch4c57.png is the function of Problem 3-41 extended so that it is 0 on the positive ch4c58.png -axis. Let ch4c59.png so that ch4c60.png is an integer because ch4c61.png . Define ch4c62.png . One has ch4c63.png and ch4c64.png . On the other boundaries, ch4c65.png and ch4c66.png . So ch4c67.png , as desired.