topology on N

nikki.arm

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.

morphism

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?

gel

Are you trying to find compactifications 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.