Finding the Right Topology on N: Help Needed!

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In summary, the person is asking for help finding a topology on the set of natural numbers with exactly k limit points. They have received a tip to find countable subsets in R and use a bijection to create the desired topology on N. They are unsure of the specific meaning or approach to this question and are asking for clarification. One possible solution is to divide N into k equivalence classes and add a point at infinity for each class to create a compactification of N.
  • #1
nikki.arm
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Hi...I'm new to the forum but I need help with the following question.

I need to find a topology on N for which there are exactly k limit points. k is a positive integer.

Tips I have received: find countable subsets in R...then a bijection will produce the needed topology on N?

Any help is greatly appreciated.
 
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  • #2
I'm not sure I understand your question. What does it mean for a topology to only have k limit points?

Presumably N is the set of natural numbers. Are you trying to consider N as a free-standing topological space, or are you trying to topologize R, and then give N the subspace topology?
 
  • #3
Are you trying to find http://en.wikipedia.org/wiki/Compactification_%28mathematics%29" of N by adding k points at infinity?

You can split N into k identical copies, according to their remainders under division by k (modulo k) and add one point at infinity for each of these equivalence classes.
 
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What is a topology?

A topology is a mathematical concept that describes the arrangement of connections between nodes in a network. It helps determine how data will flow through the network and how different devices will communicate with each other.

Why is finding the right topology important?

The right topology is crucial because it can greatly impact the performance, scalability, and reliability of a network. A well-designed topology can ensure efficient data transmission, minimize network downtime, and support the growth of the network.

What are the different types of network topologies?

The main types of network topologies are bus, ring, star, mesh, and hybrid. A bus topology has all devices connected to a single backbone cable, while a ring topology connects devices in a circular loop. In a star topology, all devices are connected to a central hub or switch. A mesh topology has multiple connections between devices, and a hybrid topology is a combination of two or more topologies.

How do I determine the right topology for my network?

The right topology will depend on various factors such as the size of the network, the number and types of devices, the desired level of redundancy, and the budget. It is essential to assess these factors and consider the pros and cons of each topology before deciding on the most suitable one for your network.

What are the common challenges in finding the right topology?

Some common challenges in finding the right topology include budget constraints, changes in network requirements, and compatibility issues between devices. It is also essential to consider scalability and future growth when choosing a topology to avoid potential issues in the long run.

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