Problem regarding the cartesian product

Reference no: EM13966652

1. Let S be any set and S∞ the set of all sequences {xn }n≥1 with xn ∈ S for all n. Let C be a subset of the Cartesian product S × S∞. Also, S × S∞ is the set of all sequences {xn }n≥0 with xn ∈ S for all n = 0, 1,... . Such a set C will be viewed as defining a sense of "convergence," so that xn →C x0 will be written in place of {xn }n≥0 ∈ C . Here are some axioms: C will be called an L-convergence if it satisfies (1) to (3) below.

(1) If xn = x for all n, then xn →C x .

(2) If xn →C x , then any subsequence xn(k) →C x .

(3) If xn →C x and xn →C y, then x = y.

If C also satisfies (4), it is called an L∗-convergence:

(4) If for every subsequence k f→ xn(k) there is a further subsequence j f→ y j := xn(k( j )) with y j →C x , then xn →C x.

(a) Prove that if T is a Hausdorff topology and C (T ) is convergence for T , then C (T ) is an L∗-convergence.

(b) Let C be any L-convergence. Let U ∈ T (C ) iff whenever xn →C x and x ∈ U , there is an m such that xn ∈ U for all n ≥ m. Prove that T (C ) is a topology.

(c) Let X be the set of all sequences {xn }n≥0 of real numbers such that for some m, xn = 0 for all n ≥ m. If y(m) = {y(m)n }n≥0 ∈ X for all m = 0, 1,... , say y(m) →C y(0) if for some k, y(m) j = y(0) j = 0 for all j ≥ k and all m, and y(m)n → y(0)n as m → ∞ for all n. Prove that →C is an L∗-convergence but that there is no metric e such that y(m) →C y(0) is equivalent to e(y(m), y(0)) → 0.

Reference no: EM13966652

Questions Cloud

What is the flux density in each of the legs : A core with three legs is shown in Figure P1-5. Its depth is 5 cm, and there are 200 turns on the leftmost leg. The relative permeability of the core can be assumed to be 1500 and constant. What flux exists in each of the three legs of the core..

Separable topological spaces : If there are infinitely many discrete Si , each having more than one point, show that their product topology is not discrete. Show that the product of countably many separable topological spaces, with product topology, is separable.

How can we sustainably use forest resources : What is an "ecosystem service?" Describe three such services provided by forests. How can we sustainably use forest resources?

Analyze the positioning of your company''s product in the u.s : The marketing report should include research on the following regarding your selected home country and the entry to U.S. market.

Problem regarding the cartesian product : Let S be any set and S∞ the set of all sequences {xn }n≥1 with xn ∈ S for all n. Let C be a subset of the Cartesian product S × S∞. Also, S × S∞ is the set of all sequences {xn }n≥0 with xn ∈ S for all n = 0, 1,... .

Linearization of a differentiable function : Linearization of a differentiable function - Write the formulas for L1(x) and L2(x) - How are the slopes of L1(x) and L2(x) related?

Summarize each of the given poem in one or two sentences : Summarize each poem in one or two sentences. What happens in the poem? Who is speaking? What is the poem's message?

Should consumers trust companies. why or why not : What are the ethical issues and obligations that come with trust once it is built?

Sequence of distinct points converges : Let X be an infinite set. Let T consist of the empty set and all complements of finite subsets of X . Show that T is a topology in which every singleton {x } is closed, but T is not metrizable. Hint: A sequence of distinct points converges to ever..

User Account

All Pages