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take_home8 - 21 Show that the Klein bottle can be obtained...

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Unformatted text preview: 21] Show that the Klein bottle can be obtained by pasting two Mbbius strips along their boundaries. 22] Let or be an irrational number, and let an 2 eh”. Show that the subgroup of SI: n=oo {3: }n.=— car:- is dense in Si. (Hint: pigeonhole principle — if n + 1 objects are placed in n: pigeonholes, then boo of the objects must wind up in the same pigeonhole.) 23] Show that the rational numbers can be arranged in a two-sided sequence {TE 2:? so that Fair”) 2 Tn+1 is a homeomorphism. (Hints: previous problemr stereographio projection and homework problem 3.4.} ...
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