Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Is my reasoning correct?

Problem:

Let $(Z,\tau)$ be the cofinite topology on $Z$. Find the limit points of the sets:

  1. $A = \{1,2,\dots,10\}$
  2. $E$, the even integers

My solution:

  1. $A$ is closed (finite), so it contains its limit points. Let $x$ in $A$, then $\{x\} \cup (Z-A)$ is a neighborhood of $x$ containing no points of $A$ other than $x$, so $A$ has no limit points.

  2. Every open neighborhood of a point $x$ in $Z$ must contain infinitely many points of $E$ for the complement to be finite. So every point in $Z$ is a limit point of $E$ (i.e., $E$ is dense in $Z$).

share|improve this question
1  
Yep. ${}{}{}{}$ –  anon Jul 24 '13 at 1:55
    
Thanks${}{}{}{}$ –  saadtaame Jul 24 '13 at 1:59
    
Do you see how you can generalize both cases? The generalization allows you to determine the limit points of any set $S\subseteq \bf Z$. For that matter you can put the cofinite topology on any countably infinite set $X$ and get the same results. –  Karl Kronenfeld Jul 24 '13 at 2:02
    
Hmm, nice observation! I think it has to do with finiteness. Right? –  saadtaame Jul 24 '13 at 2:06
    
Right, you can make that an answer if you want--it's perfectly fine to answer your own questions. –  Karl Kronenfeld Jul 24 '13 at 2:07

1 Answer 1

The proof actually applies to arbitrary sets. If $A$ is finite, the open neighborhood $\{x\}\cup(Z-A)$ contains no points of $A$ other than $x$ ($\forall x\in A$) thus $A$ has no limit points. If $A$ is infinite, then every open neighborhood of $A$ must contain infinitely many points of $A$ for the complement to be finite, so every point of $Z$ is a limit point of $A$. A subset of $Z$ is either dense or has no limit points.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.