Convergence in the product and box topology

In summary, the conversation is about a sequence of functions and whether it converges to a specific function in the cartesian product and box topologies. The person asking for help is encouraged to first try to understand the relevant concepts before seeking further assistance.
  • #1
calvin22
1
0
Hi. Can I have some help in answering the following questions? Thank you.

Let {f_n} be a sequence of functions from N(set of natural numbers) to R(real nos.) where
f_n (s)=1/n if 1<=s<=n
f_n (s)=0 if s>n.
Define f:N to R by f(s)=0 for every s>=1.
a) Does {f_n} (n=1 to inf) converge to f in the R^N (cartesian product) endowed with the product topology?
b) when endowed with the box topology?

Thanks again.
 
Last edited:
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  • #2
Of course you can can get help, but first tell us what you have tried. The least you can do is trying to come up with some relevant facts about the box and product topologies (such as their definitions, what it means for a sequence to converge in them, and the like).
 

1. What is convergence in the product and box topology?

Convergence in the product and box topology refers to the behavior of a sequence of points in a product space or a box space. It describes how the sequence of points approaches a limit point in the space.

2. How does convergence differ in the product and box topology?

In the product topology, a sequence of points converges if and only if each component of the sequence converges in its respective space. In the box topology, a sequence of points converges if and only if the sequence of components converges in each respective space.

3. Can a sequence converge in one topology but not the other?

Yes, a sequence can converge in one topology but not the other. For example, a sequence of points can converge in the product topology but not in the box topology if the components of the sequence do not converge in each respective space.

4. How does the choice of topology affect the convergence of a sequence?

The choice of topology can greatly affect the convergence of a sequence. In general, the product topology is more restrictive and requires that each component of the sequence converges, while the box topology is more permissive and only requires that the sequence of components converges. This means that a sequence may converge in the box topology but not in the product topology.

5. What are some applications of convergence in the product and box topology?

Convergence in the product and box topology is important in many areas of mathematics, especially in functional analysis and topology. It is also used in various fields of science and engineering such as computer science, physics, and economics, to study the behavior of sequences of data or events.

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