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Topology on N

by nikki.arm
Tags: limit points, topology
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Sep4-08, 10:43 PM
P: 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.
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Sep5-08, 02:11 AM
Sci Advisor
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P: 2,020
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?
Sep7-08, 08:14 PM
gel's Avatar
P: 532
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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