Topology on N

  1. 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.
  2. jcsd
  3. morphism

    morphism 2,020
    Science Advisor
    Homework Helper

    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?
  4. 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.
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