Metrizes the product of the di topologies

Assignment Help Basic Statistics

Reference no: EM13966688

If (Si , di ) are metric spaces for i ∈ I , where I is a finite set, then on the Cartesian product S = ITi ∈I Si let d(x, y) = },i di (xi , yi ).

(a) Show that d is a metric.

(b) Show that d metrizes the product of the di topologies.

Reference no: EM13966688

Questions Cloud

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..

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.

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..

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.

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.

Advise attorney howe of his potential personal liability : Write a one- to two-page (250- to 500-word) letter advising attorney Howe of his potential personal liability in Annabelle Adams's legal malpractice lawsuit against the law firm and its individual partners.

Which personal characteristics of effective counselors : Which personal characteristics of effective counselors from Ch. 2 would be most helpful in this situation? Which of the issues faced by beginning therapists from Ch. 2 might become an issue for you in this specific case (at least 2 issues) and why

Compact metric space : If (K, d) is a compact metric space and u ∈ K , show that for any finite M and 0 α ≤ 1, { f ∈ Lip(α, M ): | f (u)|≤ M } is compact for dsup. 2. If S = [0, 1] with its usual metric and α > 1, show that Lip(α, 1) contains only constant functions...

What are the reluctances of the central and right legs : A core with three legs is shown in Figure. Its depth is 8 cm, and there are 400 turns on the center leg. The remaining dimensions are shown in the figure. The core is composed of a steel having the magnetization curve shown in Figure. Answer the ..

User Account

All Pages