How Can I Define a Topology on N with Exactly k Limit Points?

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The discussion focuses on defining a topology on the set of natural numbers (N) that contains exactly k limit points, where k is a positive integer. Participants suggest utilizing countable subsets of the real numbers (R) and establishing a bijection to create the desired topology. Additionally, the concept of compactification is introduced, where N can be modified by adding k points at infinity. A method is proposed to partition N into k identical copies based on their remainders when divided by k, facilitating the addition of limit points.

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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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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?
 
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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