Reference no: EM13966692
1. Show that for k = 2, 3,... , there is a continuous function f (k) from [0, 1] onto the unit cube [0, 1]k in Rk . Hint: Let f (2)(t ) := (g(t ), h(t )) := f (t ) for 0 ≤ t ≤ 1 from Problem 9. For any (x, y, z) ∈ [0, 1]3, there are t and u in [0, 1] with f (u) = (y, z) and f (t ) = (x, u), so f (3)(t ) := (g(t ), g(h(t )), h(h(t ))) = (x, y, z). Iterate this construction.
2. Show that there is a continuous function from [0, 1] onto Tin≥1[0, 1]n , a countable product of copies of [0, 1], with product topology. Hint: Take the sequence f (k) as in Problem 10. Let Fk (t )n := f (k)(t )n for n ≤ k, 0 for n > k. Show that Fk converge to the desired function as k → ∞.
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Closure of a nowhere dense
: 1. Show that the closure of a nowhere dense set is nowhere dense. 2. Let (S, d) and (V, e) be two metric spaces. On the Cartesian product S × V take the metric ρ((x, u), (y, v)) = d(x, y) + e(u, v).
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What actions can the fed take to conduct monetary policy
: What actions can the Fed take to conduct monetary policy? What are some of the effects we would expect to see from contractionary or expansionary monetary policy?
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Where does your target market fit in the fashion cycle
: The next step in developing a promotion campaign is to understand who your customer is. Conduct research and prepare a report describing your target market in terms of:
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Compact hausdorff space
: Let K be a compact Hausdorff space and suppose for some k there are k continuous functions f1,..., fk from K into R such that x i→ ( f1(x ),..., fk (x )) is one-to-one from K into Rk .
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Countable product of copies
: Show that there is a continuous function from [0, 1] onto Tin≥1[0, 1]n , a countable product of copies of [0, 1], with product topology. Hint: Take the sequence f (k) as in Problem 10. Let Fk (t )n := f (k)(t )n for n ≤ k, 0 for n > k. Show that F..
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Problem regarding the peano curves
: Show that there is a continuous function f from the unit interval [0, 1] onto the unit square [0, 1] × [0, 1]. Hints: Let f be the limit of a sequence of functions fn which will be piecewise linear.
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Problem regarding the topological space
: A function f from a topological space (S, T ) into a metric space (Y, d) is called bounded iff its range is bounded. Let Cb (S, Y, d) be the set of all bounded, continuous functions from S into Y . For f and g in Cb (S, Y, d) let dsup( f, g) := su..
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Discuss ways firms establish barriers to entry and explain
: Discuss ways firms establish barriers to entry and explain how they benefit firms but not consumers.
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Metrizes the product of the di topologies
: (a) Show that d is a metric. (b) Show that d metrizes the product of the di topologies. (c) Show that (S, d) is complete if and only if all the (Si , di ) are complete.
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