Limit points in topology Let $X=[-1,0)\cup (0,1]$ of the real line $\mathbb{R}$. What are the limit points of $X$.
I would think that it is $(0,1)\cup(0,1)$? Am I correct?

If you're talking about the set of limit points of $X$ in $\Bbb R,$ then note that if $-1\le x\le 1,$ and if $\epsilon>0,$ then $X\cap(x-\epsilon,x+\epsilon)$ contains more than one point. Hence, every $x\in[-1,1]$ is a limit point of $X$. On the other hand, if $x<-1$ or $x>1,$ then there is some $\epsilon>0$ such that $X\cap(x-\epsilon,x+\epsilon)=\emptyset.$ Hence, only the points of $[-1,1]$ are the limit points of $X$ in $\Bbb R$.
If you're talking about the set of limit points of $X$ in $X,$ then that would just be $X\cap[-1,1]=X$. Another way to show this is to show that there is no point $x\in X$ such that $\{x\}$ is relatively open in $X$--that is, such that $\{x\}=X\cap U$ for some open subset $U$ of $\Bbb R$.
