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# hw8 - F to the boundary of the square 3 Denote by B 2 the...

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Math W4051: Problem Set 8 due Wednesday, November 14 Reading: Munkres, Ch.9 § 55-58. 1. Munkres, Ch.9 § 55, exercises 2, 4(a), 4(b), 4(c) and 4(d) on p. 353. 2. Let f, g : [0 , 1] [0 , 1] 2 be two continuous paths in a square such that f (0) = (0 , 0) , f (1) = (1 , 1) , g (0) = (1 , 0) , and g (1) = (0 , 1) . Show that there exist s, t [0 , 1] with f ( s ) = g ( t ) . (Hint: If not, one can define a map F : [0 , 1] 2 R 2 -{ 0 } by F ( s, t ) = f ( s ) - g ( t ) . Study the restriction of
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Unformatted text preview: F to the boundary of the square.) 3. Denote by B 2 the open unit ball in R 2 . Give an example of a continuous map f : B 2 → B 2 with no ﬁxed points. Challenge (extra credit): Construct a set X ⊂ R 2 (with the standard topology) such that the closure of X is R 2 and every continuous map f : X → X has a ﬁxed point....
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