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# hw2 - y-d | = s of side length 2 s> 4(a Show that the...

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Math W4051: Problem Set 2 due Wednesday, September 19 Reading: Munkres, Ch.1, § 7, and Ch.2, § 12-17. 1. Munkres § 13, exercises 1 and 3 on p.83. 2. Munkres § 16, exercise 3 on p. 92. Note: All the spaces below are endowed with the restriction of the Euclidean metric. 3. For each of the following pairs ( X, Y ) of metric spaces, write a formula for a homeo- morphism f : X Y and for its inverse f - 1 : Y X. (You do not need to justify in detail that it is a homeomorphism. You will get full credit if you just write down correct formulas.) (a) The open intervals X = ( a, b ) and Y = ( c, d ) , as subsets of R . (b) The real line X = R and the half-line Y = (0 , ) . (c) The real line X = R and the open interval Y = (0 , 1) . (d) The circle X = { ( x, y ) R 2 : ( x - a ) 2 + ( y - b ) 2 = r 2 } of radius r > 0 , and the square Y = { ( x, y ) R 2 : max( | x - c
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Unformatted text preview: y-d | ) = s } of side length 2 s > . 4. (a) Show that the intervals (0 , 1) and (0 , 2) are not isometric. (b) Show that R and (0 , 1) are not isometric. (c) Show that R and (0 , ∞ ) are not isometric. 5. Show that a triangle and a square in R 2 are not isometric. ( Note: You should consider an arbitrary triangle and an arbitrary square, rather than picking their sidelengths yourself. Also, when I say “triangle” and “square”, I mean their boundaries, not their interiors.) Challenge (extra credit): Show that [0 , 1] ∩ Q and (0 , 1) ∩ Q are homeomorphic. Note: We will see later in the course that [0 , 1] and (0 , 1) are not homeomorphic....
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