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.

Let $X=\{a,b\}$ with the indiscrete topology $T=\{X,\emptyset\}$. Consider the subset $A=\{a\}$. What is the limit point of $A$? I understand that limit point $x$ of $A$ means every neighborhood of $x$ intersects $A$ in some point other than $x$ itself. However, for a set $A=\{a\}$, how could I find the limit point of $A$? Thank you in advance!

share|improve this question
    
Well, you could start by listing the open sets in the topology, then looking at the ones that intersect $A$ ... there are not many open sets you need to look at. –  Old John Sep 19 '13 at 21:39
    
I think you got your definition of limit point wrong. A point $x \in X$ is a limit point of $A \subseteq X$ when all neighbourhoods of $x$ (not $A$) have non-empty intersection with $A \setminus \{x\}$. –  kahen Sep 19 '13 at 21:42
    
I think that's just a typo –  Stefan Hamcke Sep 19 '13 at 21:42
    
So the open sets in the topology is {a,b}? –  James Sep 19 '13 at 21:43
3  
No. The open sets are those in $T$. That's the definition of a topology. –  kahen Sep 19 '13 at 22:02

1 Answer 1

An open set of the topology $T$ is simply an element of $T$. For a point $x$ to be a limit point of $A$, every neighborhood of $x$ must intersect $A$ in a point other than itself. Since your space is finite we can check each point individually.

  • The point $a$ is not a limit point of $A$ because all its neighborhoods only intersect $A$ at $a$.
  • The point $b$ is a limit point of $A$ because it has only one neighborhood, and that is the whole space $X$ which intersects $A$ on a point different than $b$ (namely $a$).

There are no more points so $b$ is the only limit point of $A$.

share|improve this answer
    
Thanks for your answer! –  James Sep 20 '13 at 16:25

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.